{"id":1776,"date":"2018-01-11T20:43:15","date_gmt":"2018-01-11T20:43:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/integrals-resulting-in-inverse-trigonometric-functions\/"},"modified":"2018-02-07T20:26:51","modified_gmt":"2018-02-07T20:26:51","slug":"integrals-resulting-in-inverse-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/integrals-resulting-in-inverse-trigonometric-functions\/","title":{"raw":"5.7 Integrals Resulting in Inverse Trigonometric Functions","rendered":"5.7 Integrals Resulting in Inverse Trigonometric Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Integrate functions resulting in inverse trigonometric functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this section we focus on integrals that result in inverse trigonometric functions. We have worked with these functions before. Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/review-of-functions\/\">Functions and Graphs<\/a> that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. Also in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/defining-the-derivative\/\">Derivatives<\/a>, we developed formulas for derivatives of inverse trigonometric functions. The formulas developed there give rise directly to integration formulas involving inverse trigonometric functions.\r\n<div class=\"bc-section section\">\r\n<h1>Integrals that Result in Inverse Sine Functions<\/h1>\r\n<p id=\"fs-id1170571596362\">Let us begin this last section of the chapter with the three formulas. Along with these formulas, we use substitution to evaluate the integrals. We prove the formula for the inverse sine integral.<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Rule: Integration Formulas Resulting in Inverse Trigonometric Functions<\/h3>\r\n<p id=\"fs-id1170571528516\">The following integration formulas yield inverse trigonometric functions:<\/p>\r\n\r\n<ol id=\"fs-id1170572178178\">\r\n \t<li>\r\n<div id=\"fs-id1170572554001\" class=\"equation\">[latex]\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1170572607870\" class=\"equation\">[latex]\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}\\frac{u}{a}+C[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div class=\"equation\">[latex]\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}\\frac{u}{a}+C[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572216525\" class=\"bc-section section\">\r\n<h2>Proof<\/h2>\r\n<p id=\"fs-id1170571652151\">Let [latex]y={ \\sin }^{-1}\\frac{x}{a}.[\/latex] Then [latex]a \\sin y=x.[\/latex] Now let\u2019s use implicit differentiation. We obtain<\/p>\r\n\r\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\frac{d}{dx}(a \\sin y)&amp; =\\hfill &amp; \\frac{d}{dx}(x)\\hfill \\\\ \\\\ \\hfill a \\cos y\\frac{dy}{dx}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill \\frac{dy}{dx}&amp; =\\hfill &amp; \\frac{1}{a \\cos y}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572163738\">For [latex]-\\frac{\\pi }{2}\\le y\\le \\frac{\\pi }{2}, \\cos y\\ge 0.[\/latex] Thus, applying the Pythagorean identity [latex]{ \\sin }^{2}y+{ \\cos }^{2}y=1,[\/latex] we have [latex] \\cos y=\\sqrt{1={ \\sin }^{2}y}.[\/latex] This gives<\/p>\r\n\r\n<div id=\"fs-id1170571660097\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\frac{1}{a \\cos y}\\hfill &amp; =\\frac{1}{a\\sqrt{1-{ \\sin }^{2}y}}\\hfill \\\\ \\\\ &amp; =\\frac{1}{\\sqrt{{a}^{2}-{a}^{2}{ \\sin }^{2}y}}\\hfill \\\\ &amp; =\\frac{1}{\\sqrt{{a}^{2}-{x}^{2}}}.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1170572101851\">Then for [latex]\\text{\u2212}a\\le x\\le a,[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1170572175143\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{\\sqrt{{a}^{2}-{u}^{2}}}du={ \\sin }^{-1}(\\frac{u}{a})+C.[\/latex]<\/div>\r\n<p id=\"fs-id1170572346856\">\u25a1<\/p>\r\n\r\n<div id=\"fs-id1170571696734\" class=\"textbox examples\">\r\n<h3>Evaluating a Definite Integral Using Inverse Trigonometric Functions<\/h3>\r\n<div id=\"fs-id1170571596198\" class=\"exercise\">\r\n<div id=\"fs-id1170571678921\" class=\"textbox\">\r\n<p id=\"fs-id1170571618979\">Evaluate the definite integral [latex]{\\int }_{0}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\nWe can go directly to the formula for the antiderivative in the rule on integration formulas resulting in inverse trigonometric functions, and then evaluate the definite integral. We have\r\n<div id=\"fs-id1170572203987\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ {\\int }_{0}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}\\hfill &amp; ={ \\sin }^{-1}x{|}_{0}^{1}\\hfill \\\\ &amp; ={ \\sin }^{-1}1-{ \\sin }^{-1}0\\hfill \\\\ &amp; =\\frac{\\pi }{2}-0\\hfill \\\\ &amp; =\\frac{\\pi }{2}.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572393424\" class=\"textbox\">\r\n<p id=\"fs-id1170572212397\">Find the antiderivative of [latex]\\int \\frac{dx}{\\sqrt{1-16{x}^{2}}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572480280\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572480280\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480280\"][latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}(4x)+C[\/latex]<\/div>\r\n<div id=\"fs-id1170572232577\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572089952\">Substitute [latex]u=4x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572167984\" class=\"textbox examples\">\r\n<div id=\"fs-id1170571659174\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<h3>Finding an Antiderivative Involving an Inverse Trigonometric Function<\/h3>\r\n<p id=\"fs-id1170572570050\">Evaluate the integral [latex]\\int \\frac{dx}{\\sqrt{4-9{x}^{2}}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\nSubstitute [latex]u=3x.[\/latex] Then [latex]du=3dx[\/latex] and we have\r\n<div id=\"fs-id1170571618996\" class=\"equation unnumbered\">[latex]\\int \\frac{dx}{\\sqrt{4-9{x}^{2}}}=\\frac{1}{3}\\int \\frac{du}{\\sqrt{4-{u}^{2}}}.[\/latex]<\/div>\r\n<p id=\"fs-id1170572141145\">Applying the formula with [latex]a=2,[\/latex] we obtain<\/p>\r\n\r\n<div id=\"fs-id1170572130022\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\int \\frac{dx}{\\sqrt{4-9{x}^{2}}}\\hfill &amp; =\\frac{1}{3}\\int \\frac{du}{\\sqrt{4-{u}^{2}}}\\hfill \\\\ \\\\ &amp; =\\frac{1}{3}{ \\sin }^{-1}(\\frac{u}{2})+C\\hfill \\\\ &amp; =\\frac{1}{3}{ \\sin }^{-1}(\\frac{3x}{2})+C.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572141583\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170571698177\" class=\"exercise\">\r\n<div id=\"fs-id1170572114666\" class=\"textbox\">\r\n<p id=\"fs-id1170572209051\">Find the indefinite integral using an inverse trigonometric function and substitution for [latex]\\int \\frac{dx}{\\sqrt{9-{x}^{2}}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572557808\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572557808\"]\r\n<p id=\"fs-id1170572557808\">[latex]{ \\sin }^{-1}(\\frac{x}{3})+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572366383\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572274575\">Use the formula in the rule on integration formulas resulting in inverse trigonometric functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572546542\" class=\"textbox examples\">\r\n<h3>Evaluating a Definite Integral<\/h3>\r\n<div id=\"fs-id1170572299153\" class=\"exercise\">\r\n<div id=\"fs-id1170572233785\" class=\"textbox\">\r\n<p id=\"fs-id1170572223111\">Evaluate the definite integral [latex]{\\int }_{0}^{\\sqrt{3}\\text{\/}2}\\frac{du}{\\sqrt{1-{u}^{2}}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572150550\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572150550\"]\r\n<p id=\"fs-id1170572150550\">The format of the problem matches the inverse sine formula. Thus,<\/p>\r\n\r\n<div id=\"fs-id1170572130146\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ {\\int }_{0}^{\\sqrt{3}\\text{\/}2}\\frac{du}{\\sqrt{1-{u}^{2}}}\\hfill &amp; ={ \\sin }^{-1}u{|}_{0}^{\\sqrt{3}\\text{\/}2}\\hfill \\\\ &amp; =\\left[{ \\sin }^{-1}(\\frac{\\sqrt{3}}{2})\\right]-\\left[{ \\sin }^{-1}(0)\\right]\\hfill \\\\ &amp; =\\frac{\\pi }{3}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572223136\" class=\"bc-section section\">\r\n<h1>Integrals Resulting in Other Inverse Trigonometric Functions<\/h1>\r\n<p id=\"fs-id1170572106889\">There are six inverse trigonometric functions. However, only three integration formulas are noted in the rule on integration formulas resulting in inverse trigonometric functions because the remaining three are negative versions of the ones we use. The only difference is whether the integrand is positive or negative. Rather than memorizing three more formulas, if the integrand is negative, simply factor out \u22121 and evaluate the integral using one of the formulas already provided. To close this section, we examine one more formula: the integral resulting in the inverse tangent function.<\/p>\r\n\r\n<div id=\"fs-id1170572206423\" class=\"textbox examples\">\r\n<h3>Finding an Antiderivative Involving the Inverse Tangent Function<\/h3>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572134367\" class=\"textbox\">\r\n<p id=\"fs-id1170572563030\">Find an antiderivative of [latex]\\int \\frac{1}{1+4{x}^{2}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572449549\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572449549\"]\r\n<p id=\"fs-id1170572449549\">Comparing this problem with the formulas stated in the rule on integration formulas resulting in inverse trigonometric functions, the integrand looks similar to the formula for [latex]{ \\tan }^{-1}u+C.[\/latex] So we use substitution, letting [latex]u=2x,[\/latex] then [latex]du=2dx[\/latex] and [latex]1\\text{\/}2du=dx.[\/latex] Then, we have<\/p>\r\n\r\n<div id=\"fs-id1170572548844\" class=\"equation unnumbered\">[latex]\\frac{1}{2}\\int \\frac{1}{1+{u}^{2}}du=\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}u+C=\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(2x)+C.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572608152\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572308096\" class=\"exercise\">\r\n<div id=\"fs-id1170572130797\" class=\"textbox\">\r\n\r\nUse substitution to find the antiderivative of [latex]\\int \\frac{dx}{25+4{x}^{2}}.[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571562665\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571562665\"]\r\n<p id=\"fs-id1170571562665\">[latex]\\frac{1}{10}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{2x}{5})+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170571698924\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572286500\">Use the solving strategy from <a class=\"autogenerated-content\" href=\"#fs-id1170572206423\">(Figure)<\/a> and the rule on integration formulas resulting in inverse trigonometric functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572350769\" class=\"textbox examples\">\r\n<h3>Applying the Integration Formulas<\/h3>\r\n<div id=\"fs-id1170572230273\" class=\"exercise\">\r\n<div id=\"fs-id1170572216196\" class=\"textbox\">\r\n<p id=\"fs-id1170571619028\">Find the antiderivative of [latex]\\int \\frac{1}{9+{x}^{2}}dx.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1170572237513\">Apply the formula with [latex]a=3.[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1170572221498\" class=\"equation unnumbered\">[latex]\\int \\frac{dx}{9+{x}^{2}}=\\frac{1}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{3})+C.[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571681250\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1170572243063\" class=\"exercise\">\r\n<div id=\"fs-id1170572294817\" class=\"textbox\">\r\n<p id=\"fs-id1170572294820\">Find the antiderivative of [latex]\\int \\frac{dx}{16+{x}^{2}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571609481\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571609481\"]\r\n<p id=\"fs-id1170571609481\">[latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{4})+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572539671\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572498761\">Follow the steps in <a class=\"autogenerated-content\" href=\"#fs-id1170572350769\">(Figure)<\/a>.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571714255\" class=\"textbox examples\">\r\n<h3>Evaluating a Definite Integral<\/h3>\r\n<div id=\"fs-id1170572220235\" class=\"exercise\">\r\n<div id=\"fs-id1170572220237\" class=\"textbox\">\r\n<p id=\"fs-id1170571600573\">Evaluate the definite integral [latex]{\\int }_{\\sqrt{3}\\text{\/}3}^{\\sqrt{3}}\\frac{dx}{1+{x}^{2}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572099768\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572099768\"]\r\n<p id=\"fs-id1170572099768\">Use the formula for the inverse tangent. We have<\/p>\r\n\r\n<div id=\"fs-id1170572099771\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ {\\int }_{\\sqrt{3}\\text{\/}3}^{\\sqrt{3}}\\frac{dx}{1+{x}^{2}}\\hfill &amp; ={ \\tan }^{-1}x{|}_{\\sqrt{3}\\text{\/}3}^{\\sqrt{3}}\\hfill \\\\ &amp; =\\left[{ \\tan }^{-1}(\\sqrt{3})\\right]-\\left[{ \\tan }^{-1}(\\frac{\\sqrt{3}}{3})\\right]\\hfill \\\\ &amp; =\\frac{\\pi }{6}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572528694\" class=\"textbox exercises checkpoint\">\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572480550\" class=\"textbox\">\r\n<p id=\"fs-id1170572369374\">Evaluate the definite integral [latex]{\\int }_{0}^{2}\\frac{dx}{4+{x}^{2}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572176762\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572176762\"]\r\n<p id=\"fs-id1170572176762\">[latex]\\frac{\\pi }{8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572306207\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1170572137426\">Follow the procedures from <a class=\"autogenerated-content\" href=\"#fs-id1170571714255\">(Figure)<\/a> to solve the problem.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572380016\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1170571573238\">\r\n \t<li>Formulas for derivatives of inverse trigonometric functions developed in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/derivatives-of-exponential-and-logarithmic-functions\/\">Derivatives of Exponential and Logarithmic Functions<\/a> lead directly to integration formulas involving inverse trigonometric functions.<\/li>\r\n \t<li>Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.<\/li>\r\n \t<li>Substitution is often required to put the integrand in the correct form.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170571602206\" class=\"key-equations\">\r\n<h1>Key Equations<\/h1>\r\n<ul id=\"fs-id1170571602213\">\r\n \t<li><strong>Integrals That Produce Inverse Trigonometric Functions<\/strong>\r\n[latex]\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}(\\frac{u}{a})+C[\/latex]\r\n[latex]\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{u}{a})+C[\/latex]\r\n[latex]\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}(\\frac{u}{a})+C[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<p id=\"fs-id1170572180184\">In the following exercises, evaluate each integral in terms of an inverse trigonometric function.<\/p>\r\n\r\n<div id=\"fs-id1170572180188\" class=\"exercise\">\r\n<div id=\"fs-id1170572180190\" class=\"textbox\">\r\n<p id=\"fs-id1170572180192\">[latex]{\\int }_{0}^{\\sqrt{3}\\text{\/}2}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572331724\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572331724\"]\r\n<p id=\"fs-id1170572331724\">[latex]{ \\sin }^{-1}x{|}_{0}^{\\sqrt{3}\\text{\/}2}=\\frac{\\pi }{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572369231\" class=\"exercise\">\r\n<div id=\"fs-id1170572369233\" class=\"textbox\">\r\n<p id=\"fs-id1170572369235\">[latex]{\\int }_{-1\\text{\/}2}^{1\\text{\/}2}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571609291\" class=\"exercise\">\r\n<div id=\"fs-id1170571660171\" class=\"textbox\">\r\n<p id=\"fs-id1170571660173\">[latex]{\\int }_{\\sqrt{3}}^{1}\\frac{dx}{\\sqrt{1+{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572331752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572331752\"]\r\n<p id=\"fs-id1170572331752\">[latex]{ \\tan }^{-1}x{|}_{\\sqrt{3}}^{1}=-\\frac{\\pi }{12}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571712219\" class=\"exercise\">\r\n<div id=\"fs-id1170571712221\" class=\"textbox\">\r\n<p id=\"fs-id1170571712224\">[latex]{\\int }_{1\\text{\/}\\sqrt{3}}^{\\sqrt{3}}\\frac{dx}{1+{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572625745\" class=\"exercise\">\r\n<div id=\"fs-id1170572625748\" class=\"textbox\">\r\n<p id=\"fs-id1170572509844\">[latex]{\\int }_{1}^{\\sqrt{2}}\\frac{dx}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572480541\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480541\"]\r\n<p id=\"fs-id1170572480541\">[latex]{ \\sec }^{-1}x{|}_{1}^{\\sqrt{2}}=\\frac{\\pi }{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572560653\" class=\"exercise\">\r\n<div id=\"fs-id1170572560655\" class=\"textbox\">\r\n<p id=\"fs-id1170572560657\">[latex]{\\int }_{1}^{2\\text{\/}\\sqrt{3}}\\frac{dx}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572333181\">In the following exercises, find each indefinite integral, using appropriate substitutions.<\/p>\r\n\r\n<div id=\"fs-id1170572333184\" class=\"exercise\">\r\n<div id=\"fs-id1170572333187\" class=\"textbox\">\r\n<p id=\"fs-id1170572333189\">[latex]\\int \\frac{dx}{\\sqrt{9-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]{ \\sin }^{-1}(\\frac{x}{3})+C[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571543108\" class=\"exercise\">\r\n<div id=\"fs-id1170572163833\" class=\"textbox\">\r\n<p id=\"fs-id1170572163835\">[latex]\\int \\frac{dx}{\\sqrt{1-16{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571630356\" class=\"exercise\">\r\n<div id=\"fs-id1170571630358\" class=\"textbox\">\r\n<p id=\"fs-id1170571630360\">[latex]\\int \\frac{dx}{9+{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572614124\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572614124\"]\r\n<p id=\"fs-id1170572614124\">[latex]\\frac{1}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{3})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571719603\" class=\"exercise\">\r\n<div id=\"fs-id1170572425059\" class=\"textbox\">\r\n<p id=\"fs-id1170572425061\">[latex]\\int \\frac{dx}{25+16{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571712540\" class=\"exercise\">\r\n<div id=\"fs-id1170571712542\" class=\"textbox\">\r\n<p id=\"fs-id1170571712545\">[latex]\\int \\frac{dx}{|x|\\sqrt{{x}^{2}-9}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572559663\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572559663\"]\r\n<p id=\"fs-id1170572559663\">[latex]\\frac{1}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}(\\frac{x}{3})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170571609451\">[latex]\\int \\frac{dx}{|x|\\sqrt{4{x}^{2}-16}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572274931\" class=\"exercise\">\r\n<div id=\"fs-id1170572274933\" class=\"textbox\">\r\n<p id=\"fs-id1170572274935\">Explain the relationship [latex]\\text{\u2212}{ \\cos }^{-1}t+C=\\int \\frac{dt}{\\sqrt{1-{t}^{2}}}={ \\sin }^{-1}t+C.[\/latex] Is it true, in general, that [latex]{ \\cos }^{-1}t=\\text{\u2212}{ \\sin }^{-1}t?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571788133\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571788133\"]\r\n<p id=\"fs-id1170571788133\">[latex] \\cos (\\frac{\\pi }{2}-\\theta )= \\sin \\theta .[\/latex] So, [latex]{ \\sin }^{-1}t=\\frac{\\pi }{2}-{ \\cos }^{-1}t.[\/latex] They differ by a constant.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572563040\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170572563044\">Explain the relationship [latex]{ \\sec }^{-1}t+C=\\int \\frac{dt}{|t|\\sqrt{{t}^{2}-1}}=\\text{\u2212}{ \\csc }^{-1}t+C.[\/latex] Is it true, in general, that [latex]{ \\sec }^{-1}t=\\text{\u2212}{ \\csc }^{-1}t?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170571609260\" class=\"textbox\">\r\n<p id=\"fs-id1170571638317\">Explain what is wrong with the following integral: [latex]{\\int }_{1}^{2}\\frac{dt}{\\sqrt{1-{t}^{2}}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572624827\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572624827\"]\r\n<p id=\"fs-id1170572624827\">[latex]\\sqrt{1-{t}^{2}}[\/latex] is not defined as a real number when [latex]t&gt;1.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572163878\" class=\"exercise\">\r\n<div id=\"fs-id1170572163880\" class=\"textbox\">\r\n<p id=\"fs-id1170572163882\">Explain what is wrong with the following integral: [latex]{\\int }_{-1}^{1}\\frac{dt}{|t|\\sqrt{{t}^{2}-1}}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nIn the following exercises, solve for the antiderivative [latex]\\int f[\/latex] of [latex]f[\/latex] with [latex]C=0,[\/latex] then use a calculator to graph [latex]f[\/latex] and the antiderivative over the given interval [latex]\\left[a,b\\right].[\/latex] Identify a value of <em>C<\/em> such that adding <em>C<\/em> to the antiderivative recovers the definite integral [latex]F(x)={\\int }_{a}^{x}f(t)dt.[\/latex]\r\n<div id=\"fs-id1170571699012\" class=\"exercise\">\r\n<div id=\"fs-id1170571699014\" class=\"textbox\">\r\n<p id=\"fs-id1170571699016\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{\\sqrt{9-{x}^{2}}}dx[\/latex] over [latex]\\left[-3,3\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572274896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572274896\"]<span id=\"fs-id1170572274902\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204301\/CNX_Calc_Figure_05_07_201.jpg\" alt=\"Two graphs. The first shows the function f(x) = 1 \/ sqrt(9 \u2013 x^2). It is an upward opening curve symmetric about the y axis, crossing at (0, 1\/3). The second shows the function F(x) = arcsin(1\/3 x). It is an increasing curve going through the origin.\" \/><\/span>\r\nThe antiderivative is [latex]{ \\sin }^{-1}(\\frac{x}{3})+C.[\/latex] Taking [latex]C=\\frac{\\pi }{2}[\/latex] recovers the definite integral.[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571649965\" class=\"exercise\">\r\n<div id=\"fs-id1170571649967\" class=\"textbox\">\r\n<p id=\"fs-id1170571649969\"><strong>[T]<\/strong>[latex]\\int \\frac{9}{9+{x}^{2}}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572565396\" class=\"exercise\">\r\n<div id=\"fs-id1170572565398\" class=\"textbox\">\r\n\r\n<strong>[T]<\/strong>[latex]\\int \\frac{ \\cos x}{4+{ \\sin }^{2}x}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571678881\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571678881\"]<span id=\"fs-id1170571678884\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204305\/CNX_Calc_Figure_05_07_203.jpg\" alt=\"Two graphs. The first shows the function f(x) = cos(x) \/ (4 + sin(x)^2). It is an oscillating function over [-6, 6] with turning points at roughly (-3, -2.5), (0, .25), and (3, -2.5), where (0,.25) is a local max and the others are local mins. The second shows the function F(x) = .5 * arctan(.5*sin(x)), which also oscillates over [-6,6]. It has turning points at roughly (-4.5, .25), (-1.5, -.25), (1.5, .25), and (4.5, -.25).\" \/><\/span>\r\nThe antiderivative is [latex]\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{ \\sin x}{2})+C.[\/latex] Taking [latex]C=\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{ \\sin (6)}{2})[\/latex] recovers the definite integral.[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572368670\" class=\"exercise\">\r\n<div id=\"fs-id1170572368672\" class=\"textbox\">\r\n<p id=\"fs-id1170572368675\"><strong>[T]<\/strong>[latex]\\int \\frac{{e}^{x}}{1+{e}^{2x}}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170571769590\">In the following exercises, compute the antiderivative using appropriate substitutions.<\/p>\r\n\r\n<div id=\"fs-id1170571769593\" class=\"exercise\">\r\n<div id=\"fs-id1170571769595\" class=\"textbox\">\r\n<p id=\"fs-id1170571769597\">[latex]\\int \\frac{{ \\sin }^{-1}tdt}{\\sqrt{1-{t}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572611876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572611876\"]\r\n<p id=\"fs-id1170572611876\">[latex]\\frac{1}{2}{({ \\sin }^{-1}t)}^{2}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571636151\" class=\"exercise\">\r\n<div id=\"fs-id1170571636153\" class=\"textbox\">\r\n<p id=\"fs-id1170571636155\">[latex]\\int \\frac{dt}{{ \\sin }^{-1}t\\sqrt{1-{t}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572628413\" class=\"exercise\">\r\n<div id=\"fs-id1170572628415\" class=\"textbox\">\r\n\r\n[latex]\\int \\frac{{ \\tan }^{-1}(2t)}{1+4{t}^{2}}dt[\/latex]\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572628467\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572628467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572628467\"][latex]\\frac{1}{4}{({ \\tan }^{-1}(2t))}^{2}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572331872\" class=\"exercise\">\r\n<div id=\"fs-id1170572331874\" class=\"textbox\">\r\n<p id=\"fs-id1170572331876\">[latex]\\int \\frac{t{ \\tan }^{-1}({t}^{2})}{1+{t}^{4}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572558294\" class=\"exercise\">\r\n<div id=\"fs-id1170572558296\" class=\"textbox\">\r\n<p id=\"fs-id1170572558299\">[latex]\\int \\frac{{ \\sec }^{-1}(\\frac{t}{2})}{|t|\\sqrt{{t}^{2}-4}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572184332\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572184332\"]\r\n<p id=\"fs-id1170572184332\">[latex]\\frac{1}{4}({ \\sec }^{-1}{(\\frac{t}{2})}^{2})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571733959\" class=\"exercise\">\r\n<div id=\"fs-id1170571733961\" class=\"textbox\">\r\n<p id=\"fs-id1170571733963\">[latex]\\int \\frac{t{ \\sec }^{-1}({t}^{2})}{{t}^{2}\\sqrt{{t}^{4}-1}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nIn the following exercises, use a calculator to graph the antiderivative [latex]\\int f[\/latex] with [latex]C=0[\/latex] over the given interval [latex]\\left[a,b\\right].[\/latex] Approximate a value of <em>C<\/em>, if possible, such that adding <em>C<\/em> to the antiderivative gives the same value as the definite integral [latex]F(x)={\\int }_{a}^{x}f(t)dt.[\/latex]\r\n<div id=\"fs-id1170572601262\" class=\"exercise\">\r\n<div id=\"fs-id1170572601264\" class=\"textbox\">\r\n<p id=\"fs-id1170572601266\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{x\\sqrt{{x}^{2}-4}}dx[\/latex] over [latex]\\left[2,6\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571679782\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571679782\"]<span id=\"fs-id1170571679787\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204308\/CNX_Calc_Figure_05_07_205.jpg\" alt=\"A graph of the function f(x) = -.5 * arctan(2 \/ ( sqrt(x^2 \u2013 4) ) ) in quadrant four. It is an increasing concave down curve with a vertical asymptote at x=2.\" \/><\/span>\r\nThe antiderivative is [latex]\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}(\\frac{x}{2})+C.[\/latex] Taking [latex]C=0[\/latex] recovers the definite integral over [latex]\\left[2,6\\right].[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571679862\" class=\"exercise\">\r\n<div id=\"fs-id1170571679864\" class=\"textbox\">\r\n<p id=\"fs-id1170571679866\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{(2x+2)\\sqrt{x}}dx[\/latex] over [latex]\\left[0,6\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572249934\" class=\"exercise\">\r\n<div id=\"fs-id1170572249937\" class=\"textbox\">\r\n<p id=\"fs-id1170572249939\"><strong>[T]<\/strong>[latex]\\int \\frac{( \\sin x+x \\cos x)}{1+{x}^{2}{ \\sin }^{2}x}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204310\/CNX_Calc_Figure_05_07_207.jpg\" alt=\"The graph of f(x) = arctan(x sin(x)) over [-6,6]. It has five turning points at roughly (-5, -1.5), (-2,1), (0,0), (2,1), and (5,-1.5).\" \/>\r\nThe general antiderivative is [latex]{ \\tan }^{-1}(x \\sin x)+C.[\/latex] Taking [latex]C=\\text{\u2212}{ \\tan }^{-1}(6 \\sin (6))[\/latex] recovers the definite integral.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571810858\" class=\"exercise\">\r\n<div id=\"fs-id1170571810860\" class=\"textbox\">\r\n<p id=\"fs-id1170571810862\"><strong>[T]<\/strong>[latex]\\int \\frac{2{e}^{-2x}}{\\sqrt{1-{e}^{-4x}}}dx[\/latex] over [latex]\\left[0,2\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571697193\" class=\"exercise\">\r\n<div id=\"fs-id1170571697196\" class=\"textbox\">\r\n<p id=\"fs-id1170571697198\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{x+x{\\text{ln}}^{2}x}[\/latex] over [latex]\\left[0,2\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n<span id=\"fs-id1170571539140\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204313\/CNX_Calc_Figure_05_07_209.jpg\" alt=\"A graph of the function f(x) = arctan(ln(x)) over (0, 2]. It is an increasing curve with x-intercept at (1,0).\" \/><\/span>\r\nThe general antiderivative is [latex]{ \\tan }^{-1}(\\text{ln}x)+C.[\/latex] Taking [latex]C=\\frac{\\pi }{2}={ \\tan }^{-1}\\infty [\/latex] recovers the definite integral.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571539209\" class=\"exercise\">\r\n<div id=\"fs-id1170571543184\" class=\"textbox\">\r\n<p id=\"fs-id1170571543187\"><strong>[T]<\/strong>[latex]\\int \\frac{{ \\sin }^{-1}x}{\\sqrt{1-{x}^{2}}}[\/latex] over [latex]\\left[-1,1\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572456361\">In the following exercises, compute each integral using appropriate substitutions.<\/p>\r\n\r\n<div id=\"fs-id1170572456364\" class=\"exercise\">\r\n<div id=\"fs-id1170572456366\" class=\"textbox\">\r\n\r\n[latex]\\int \\frac{{e}^{x}}{\\sqrt{1-{e}^{2t}}}dt[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572456411\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572456411\"]\r\n<p id=\"fs-id1170572456411\">[latex]{ \\sin }^{-1}({e}^{t})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571580944\" class=\"exercise\">\r\n<div id=\"fs-id1170571580946\" class=\"textbox\">\r\n\r\n[latex]\\int \\frac{{e}^{t}}{1+{e}^{2t}}dt[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572396478\" class=\"exercise\">\r\n<div id=\"fs-id1170572396480\" class=\"textbox\">\r\n<p id=\"fs-id1170572396482\">[latex]\\int \\frac{dt}{t\\sqrt{1-{\\text{ln}}^{2}t}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572396521\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572396521\"]\r\n<p id=\"fs-id1170572396521\">[latex]{ \\sin }^{-1}(\\text{ln}t)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572396552\" class=\"exercise\">\r\n<div id=\"fs-id1170572396554\" class=\"textbox\">\r\n<p id=\"fs-id1170572396556\">[latex]\\int \\frac{dt}{t(1+{\\text{ln}}^{2}t)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572218636\" class=\"exercise\">\r\n<div id=\"fs-id1170572218638\" class=\"textbox\">\r\n<p id=\"fs-id1170572218640\">[latex]\\int \\frac{{ \\cos }^{-1}(2t)}{\\sqrt{1-4{t}^{2}}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572386126\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572386126\"]\r\n<p id=\"fs-id1170572386126\">[latex]-\\frac{1}{2}{({ \\cos }^{-1}(2t))}^{2}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572386173\" class=\"exercise\">\r\n<div id=\"fs-id1170572386175\" class=\"textbox\">\r\n<p id=\"fs-id1170572386177\">[latex]\\int \\frac{{e}^{t}{ \\cos }^{-1}({e}^{t})}{\\sqrt{1-{e}^{2t}}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572643268\">In the following exercises, compute each definite integral.<\/p>\r\n\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170572643276\">[latex]{\\int }_{0}^{1\\text{\/}2}\\frac{ \\tan ({ \\sin }^{-1}t)}{\\sqrt{1-{t}^{2}}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n\r\n[latex]\\frac{1}{2}\\text{ln}(\\frac{4}{3})[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572373740\" class=\"exercise\">\r\n<div id=\"fs-id1170572373742\" class=\"textbox\">\r\n<p id=\"fs-id1170572373744\">[latex]{\\int }_{1\\text{\/}4}^{1\\text{\/}2}\\frac{ \\tan ({ \\cos }^{-1}t)}{\\sqrt{1-{t}^{2}}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571710706\" class=\"exercise\">\r\n<div id=\"fs-id1170571710708\" class=\"textbox\">\r\n\r\n[latex]{\\int }_{0}^{1\\text{\/}2}\\frac{ \\sin ({ \\tan }^{-1}t)}{1+{t}^{2}}dt[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572399013\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572399013\"]\r\n<p id=\"fs-id1170572399013\">[latex]1-\\frac{2}{\\sqrt{5}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572399032\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1170572399036\">[latex]{\\int }_{0}^{1\\text{\/}2}\\frac{ \\cos ({ \\tan }^{-1}t)}{1+{t}^{2}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572415154\" class=\"textbox\">\r\n<p id=\"fs-id1170572415156\">For [latex]A&gt;0,[\/latex] compute [latex]I(A)={\\int }_{\\text{\u2212}A}^{A}\\frac{dt}{1+{t}^{2}}[\/latex] and evaluate [latex]\\underset{a\\to \\infty }{\\text{lim}}I(A),[\/latex] the area under the graph of [latex]\\frac{1}{1+{t}^{2}}[\/latex] on [latex]\\left[\\text{\u2212}\\infty ,\\infty \\right].[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572168753\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572168753\"]\r\n<p id=\"fs-id1170572168753\">[latex]2{ \\tan }^{-1}(A)\\to \\pi [\/latex] as [latex]A\\to \\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1170572629162\" class=\"textbox\">\r\n<p id=\"fs-id1170572629164\">For [latex]1&lt;B&lt;\\infty ,[\/latex] compute [latex]I(B)={\\int }_{1}^{B}\\frac{dt}{t\\sqrt{{t}^{2}-1}}[\/latex] and evaluate [latex]\\underset{B\\to \\infty }{\\text{lim}}I(B),[\/latex] the area under the graph of [latex]\\frac{1}{t\\sqrt{{t}^{2}-1}}[\/latex] over [latex]\\left[1,\\infty ).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572569940\" class=\"exercise\">\r\n<div id=\"fs-id1170572569942\" class=\"textbox\">\r\n<p id=\"fs-id1170572569944\">Use the substitution [latex]u=\\sqrt{2} \\cot x[\/latex] and the identity [latex]1+{ \\cot }^{2}x={ \\csc }^{2}x[\/latex] to evaluate [latex]\\int \\frac{dx}{1+{ \\cos }^{2}x}.[\/latex] (<em>Hint:<\/em> Multiply the top and bottom of the integrand by [latex]{ \\csc }^{2}x.[\/latex])<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170572378997\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572378997\"]\r\n<p id=\"fs-id1170572378997\">Using the hint, one has [latex]\\int \\frac{{ \\csc }^{2}x}{{ \\csc }^{2}x+{ \\cot }^{2}x}dx=\\int \\frac{{ \\csc }^{2}x}{1+2{ \\cot }^{2}x}dx.[\/latex] Set [latex]u=\\sqrt{2} \\cot x.[\/latex] Then, [latex]du=\\text{\u2212}\\sqrt{2}{ \\csc }^{2}x[\/latex] and the integral is [latex]-\\frac{1}{\\sqrt{2}}\\int \\frac{du}{1+{u}^{2}}=-\\frac{1}{\\sqrt{2}}{ \\tan }^{-1}u+C=\\frac{1}{\\sqrt{2}}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\sqrt{2} \\cot x)+C.[\/latex] If one uses the identity [latex]{ \\tan }^{-1}s+{ \\tan }^{-1}(\\frac{1}{s})=\\frac{\\pi }{2},[\/latex] then this can also be written [latex]\\frac{1}{\\sqrt{2}}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{ \\tan x}{\\sqrt{2}})+C.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572310020\" class=\"exercise\">\r\n<div id=\"fs-id1170572310022\" class=\"textbox\">\r\n<p id=\"fs-id1170572310024\"><strong>[T]<\/strong> Approximate the points at which the graphs of [latex]f(x)=2{x}^{2}-1[\/latex] and [latex]g(x)={(1+4{x}^{2})}^{-3\\text{\/}2}[\/latex] intersect, and approximate the area between their graphs accurate to three decimal places.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571613771\" class=\"exercise\">\r\n<div id=\"fs-id1170571613774\" class=\"textbox\">\r\n<p id=\"fs-id1170571613776\">47. <strong>[T]<\/strong> Approximate the points at which the graphs of [latex]f(x)={x}^{2}-1[\/latex] and [latex]f(x)={x}^{2}-1[\/latex] intersect, and approximate the area between their graphs accurate to three decimal places.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1170571661768\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571661768\"]\r\n<p id=\"fs-id1170571661768\">[latex]x\\approx \u00b11.13525.[\/latex] The left endpoint estimate with [latex]N=100[\/latex] is 2.796 and these decimals persist for [latex]N=500.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571661806\" class=\"exercise\">\r\n<div id=\"fs-id1170571661808\" class=\"textbox\">\r\n<p id=\"fs-id1170571661810\">Use the following graph to prove that [latex]{\\int }_{0}^{x}\\sqrt{1-{t}^{2}}dt=\\frac{1}{2}x\\sqrt{1-{x}^{2}}+\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}x.[\/latex]<\/p>\r\n<span id=\"fs-id1170572331317\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204315\/CNX_Calc_Figure_05_07_211.jpg\" alt=\"A diagram containing two shapes, a wedge from a circle shaded in blue on top of a triangle shaded in brown. The triangle\u2019s hypotenuse is one of the radii edges of the wedge of the circle and is 1 unit long. There is a dotted red line forming a rectangle out of part of the wedge and the triangle, with the hypotenuse of the triangle as the diagonal of the rectangle. The curve of the circle is described by the equation sqrt(1-x^2).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572346977\" class=\"review-exercises\"><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Integrate functions resulting in inverse trigonometric functions<\/li>\n<\/ul>\n<\/div>\n<p>In this section we focus on integrals that result in inverse trigonometric functions. We have worked with these functions before. Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/review-of-functions\/\">Functions and Graphs<\/a> that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. Also in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/defining-the-derivative\/\">Derivatives<\/a>, we developed formulas for derivatives of inverse trigonometric functions. The formulas developed there give rise directly to integration formulas involving inverse trigonometric functions.<\/p>\n<div class=\"bc-section section\">\n<h1>Integrals that Result in Inverse Sine Functions<\/h1>\n<p id=\"fs-id1170571596362\">Let us begin this last section of the chapter with the three formulas. Along with these formulas, we use substitution to evaluate the integrals. We prove the formula for the inverse sine integral.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Rule: Integration Formulas Resulting in Inverse Trigonometric Functions<\/h3>\n<p id=\"fs-id1170571528516\">The following integration formulas yield inverse trigonometric functions:<\/p>\n<ol id=\"fs-id1170572178178\">\n<li>\n<div id=\"fs-id1170572554001\" class=\"equation\">[latex]\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1170572607870\" class=\"equation\">[latex]\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}\\frac{u}{a}+C[\/latex]<\/div>\n<\/li>\n<li>\n<div class=\"equation\">[latex]\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}\\frac{u}{a}+C[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572216525\" class=\"bc-section section\">\n<h2>Proof<\/h2>\n<p id=\"fs-id1170571652151\">Let [latex]y={ \\sin }^{-1}\\frac{x}{a}.[\/latex] Then [latex]a \\sin y=x.[\/latex] Now let\u2019s use implicit differentiation. We obtain<\/p>\n<div class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\frac{d}{dx}(a \\sin y)& =\\hfill & \\frac{d}{dx}(x)\\hfill \\\\ \\\\ \\hfill a \\cos y\\frac{dy}{dx}& =\\hfill & 1\\hfill \\\\ \\hfill \\frac{dy}{dx}& =\\hfill & \\frac{1}{a \\cos y}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572163738\">For [latex]-\\frac{\\pi }{2}\\le y\\le \\frac{\\pi }{2}, \\cos y\\ge 0.[\/latex] Thus, applying the Pythagorean identity [latex]{ \\sin }^{2}y+{ \\cos }^{2}y=1,[\/latex] we have [latex]\\cos y=\\sqrt{1={ \\sin }^{2}y}.[\/latex] This gives<\/p>\n<div id=\"fs-id1170571660097\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\frac{1}{a \\cos y}\\hfill & =\\frac{1}{a\\sqrt{1-{ \\sin }^{2}y}}\\hfill \\\\ \\\\ & =\\frac{1}{\\sqrt{{a}^{2}-{a}^{2}{ \\sin }^{2}y}}\\hfill \\\\ & =\\frac{1}{\\sqrt{{a}^{2}-{x}^{2}}}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1170572101851\">Then for [latex]\\text{\u2212}a\\le x\\le a,[\/latex] we have<\/p>\n<div id=\"fs-id1170572175143\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{\\sqrt{{a}^{2}-{u}^{2}}}du={ \\sin }^{-1}(\\frac{u}{a})+C.[\/latex]<\/div>\n<p id=\"fs-id1170572346856\">\u25a1<\/p>\n<div id=\"fs-id1170571696734\" class=\"textbox examples\">\n<h3>Evaluating a Definite Integral Using Inverse Trigonometric Functions<\/h3>\n<div id=\"fs-id1170571596198\" class=\"exercise\">\n<div id=\"fs-id1170571678921\" class=\"textbox\">\n<p id=\"fs-id1170571618979\">Evaluate the definite integral [latex]{\\int }_{0}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}.[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>We can go directly to the formula for the antiderivative in the rule on integration formulas resulting in inverse trigonometric functions, and then evaluate the definite integral. We have<\/p>\n<div id=\"fs-id1170572203987\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ {\\int }_{0}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}\\hfill & ={ \\sin }^{-1}x{|}_{0}^{1}\\hfill \\\\ & ={ \\sin }^{-1}1-{ \\sin }^{-1}0\\hfill \\\\ & =\\frac{\\pi }{2}-0\\hfill \\\\ & =\\frac{\\pi }{2}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div id=\"fs-id1170572393424\" class=\"textbox\">\n<p id=\"fs-id1170572212397\">Find the antiderivative of [latex]\\int \\frac{dx}{\\sqrt{1-16{x}^{2}}}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170572480280\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572480280\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572480280\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}(4x)+C[\/latex]<\/div>\n<div id=\"fs-id1170572232577\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572089952\">Substitute [latex]u=4x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572167984\" class=\"textbox examples\">\n<div id=\"fs-id1170571659174\" class=\"exercise\">\n<div class=\"textbox\">\n<h3>Finding an Antiderivative Involving an Inverse Trigonometric Function<\/h3>\n<p id=\"fs-id1170572570050\">Evaluate the integral [latex]\\int \\frac{dx}{\\sqrt{4-9{x}^{2}}}.[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>Substitute [latex]u=3x.[\/latex] Then [latex]du=3dx[\/latex] and we have<\/p>\n<div id=\"fs-id1170571618996\" class=\"equation unnumbered\">[latex]\\int \\frac{dx}{\\sqrt{4-9{x}^{2}}}=\\frac{1}{3}\\int \\frac{du}{\\sqrt{4-{u}^{2}}}.[\/latex]<\/div>\n<p id=\"fs-id1170572141145\">Applying the formula with [latex]a=2,[\/latex] we obtain<\/p>\n<div id=\"fs-id1170572130022\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\int \\frac{dx}{\\sqrt{4-9{x}^{2}}}\\hfill & =\\frac{1}{3}\\int \\frac{du}{\\sqrt{4-{u}^{2}}}\\hfill \\\\ \\\\ & =\\frac{1}{3}{ \\sin }^{-1}(\\frac{u}{2})+C\\hfill \\\\ & =\\frac{1}{3}{ \\sin }^{-1}(\\frac{3x}{2})+C.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572141583\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170571698177\" class=\"exercise\">\n<div id=\"fs-id1170572114666\" class=\"textbox\">\n<p id=\"fs-id1170572209051\">Find the indefinite integral using an inverse trigonometric function and substitution for [latex]\\int \\frac{dx}{\\sqrt{9-{x}^{2}}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572557808\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572557808\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572557808\">[latex]{ \\sin }^{-1}(\\frac{x}{3})+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170572366383\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572274575\">Use the formula in the rule on integration formulas resulting in inverse trigonometric functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572546542\" class=\"textbox examples\">\n<h3>Evaluating a Definite Integral<\/h3>\n<div id=\"fs-id1170572299153\" class=\"exercise\">\n<div id=\"fs-id1170572233785\" class=\"textbox\">\n<p id=\"fs-id1170572223111\">Evaluate the definite integral [latex]{\\int }_{0}^{\\sqrt{3}\\text{\/}2}\\frac{du}{\\sqrt{1-{u}^{2}}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572150550\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572150550\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572150550\">The format of the problem matches the inverse sine formula. Thus,<\/p>\n<div id=\"fs-id1170572130146\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ {\\int }_{0}^{\\sqrt{3}\\text{\/}2}\\frac{du}{\\sqrt{1-{u}^{2}}}\\hfill & ={ \\sin }^{-1}u{|}_{0}^{\\sqrt{3}\\text{\/}2}\\hfill \\\\ & =\\left[{ \\sin }^{-1}(\\frac{\\sqrt{3}}{2})\\right]-\\left[{ \\sin }^{-1}(0)\\right]\\hfill \\\\ & =\\frac{\\pi }{3}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572223136\" class=\"bc-section section\">\n<h1>Integrals Resulting in Other Inverse Trigonometric Functions<\/h1>\n<p id=\"fs-id1170572106889\">There are six inverse trigonometric functions. However, only three integration formulas are noted in the rule on integration formulas resulting in inverse trigonometric functions because the remaining three are negative versions of the ones we use. The only difference is whether the integrand is positive or negative. Rather than memorizing three more formulas, if the integrand is negative, simply factor out \u22121 and evaluate the integral using one of the formulas already provided. To close this section, we examine one more formula: the integral resulting in the inverse tangent function.<\/p>\n<div id=\"fs-id1170572206423\" class=\"textbox examples\">\n<h3>Finding an Antiderivative Involving the Inverse Tangent Function<\/h3>\n<div class=\"exercise\">\n<div id=\"fs-id1170572134367\" class=\"textbox\">\n<p id=\"fs-id1170572563030\">Find an antiderivative of [latex]\\int \\frac{1}{1+4{x}^{2}}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572449549\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572449549\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572449549\">Comparing this problem with the formulas stated in the rule on integration formulas resulting in inverse trigonometric functions, the integrand looks similar to the formula for [latex]{ \\tan }^{-1}u+C.[\/latex] So we use substitution, letting [latex]u=2x,[\/latex] then [latex]du=2dx[\/latex] and [latex]1\\text{\/}2du=dx.[\/latex] Then, we have<\/p>\n<div id=\"fs-id1170572548844\" class=\"equation unnumbered\">[latex]\\frac{1}{2}\\int \\frac{1}{1+{u}^{2}}du=\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}u+C=\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(2x)+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572608152\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572308096\" class=\"exercise\">\n<div id=\"fs-id1170572130797\" class=\"textbox\">\n<p>Use substitution to find the antiderivative of [latex]\\int \\frac{dx}{25+4{x}^{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571562665\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571562665\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571562665\">[latex]\\frac{1}{10}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{2x}{5})+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170571698924\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572286500\">Use the solving strategy from <a class=\"autogenerated-content\" href=\"#fs-id1170572206423\">(Figure)<\/a> and the rule on integration formulas resulting in inverse trigonometric functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572350769\" class=\"textbox examples\">\n<h3>Applying the Integration Formulas<\/h3>\n<div id=\"fs-id1170572230273\" class=\"exercise\">\n<div id=\"fs-id1170572216196\" class=\"textbox\">\n<p id=\"fs-id1170571619028\">Find the antiderivative of [latex]\\int \\frac{1}{9+{x}^{2}}dx.[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1170572237513\">Apply the formula with [latex]a=3.[\/latex] Then,<\/p>\n<div id=\"fs-id1170572221498\" class=\"equation unnumbered\">[latex]\\int \\frac{dx}{9+{x}^{2}}=\\frac{1}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{3})+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571681250\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1170572243063\" class=\"exercise\">\n<div id=\"fs-id1170572294817\" class=\"textbox\">\n<p id=\"fs-id1170572294820\">Find the antiderivative of [latex]\\int \\frac{dx}{16+{x}^{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571609481\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571609481\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571609481\">[latex]\\frac{1}{4}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{4})+C[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170572539671\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572498761\">Follow the steps in <a class=\"autogenerated-content\" href=\"#fs-id1170572350769\">(Figure)<\/a>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571714255\" class=\"textbox examples\">\n<h3>Evaluating a Definite Integral<\/h3>\n<div id=\"fs-id1170572220235\" class=\"exercise\">\n<div id=\"fs-id1170572220237\" class=\"textbox\">\n<p id=\"fs-id1170571600573\">Evaluate the definite integral [latex]{\\int }_{\\sqrt{3}\\text{\/}3}^{\\sqrt{3}}\\frac{dx}{1+{x}^{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572099768\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572099768\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572099768\">Use the formula for the inverse tangent. We have<\/p>\n<div id=\"fs-id1170572099771\" class=\"equation unnumbered\">[latex]\\begin{array}{}\\\\ \\\\ {\\int }_{\\sqrt{3}\\text{\/}3}^{\\sqrt{3}}\\frac{dx}{1+{x}^{2}}\\hfill & ={ \\tan }^{-1}x{|}_{\\sqrt{3}\\text{\/}3}^{\\sqrt{3}}\\hfill \\\\ & =\\left[{ \\tan }^{-1}(\\sqrt{3})\\right]-\\left[{ \\tan }^{-1}(\\frac{\\sqrt{3}}{3})\\right]\\hfill \\\\ & =\\frac{\\pi }{6}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572528694\" class=\"textbox exercises checkpoint\">\n<div class=\"exercise\">\n<div id=\"fs-id1170572480550\" class=\"textbox\">\n<p id=\"fs-id1170572369374\">Evaluate the definite integral [latex]{\\int }_{0}^{2}\\frac{dx}{4+{x}^{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572176762\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572176762\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572176762\">[latex]\\frac{\\pi }{8}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170572306207\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1170572137426\">Follow the procedures from <a class=\"autogenerated-content\" href=\"#fs-id1170571714255\">(Figure)<\/a> to solve the problem.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572380016\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1170571573238\">\n<li>Formulas for derivatives of inverse trigonometric functions developed in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/derivatives-of-exponential-and-logarithmic-functions\/\">Derivatives of Exponential and Logarithmic Functions<\/a> lead directly to integration formulas involving inverse trigonometric functions.<\/li>\n<li>Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.<\/li>\n<li>Substitution is often required to put the integrand in the correct form.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170571602206\" class=\"key-equations\">\n<h1>Key Equations<\/h1>\n<ul id=\"fs-id1170571602213\">\n<li><strong>Integrals That Produce Inverse Trigonometric Functions<\/strong><br \/>\n[latex]\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}(\\frac{u}{a})+C[\/latex]<br \/>\n[latex]\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{u}{a})+C[\/latex]<br \/>\n[latex]\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}(\\frac{u}{a})+C[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<p id=\"fs-id1170572180184\">In the following exercises, evaluate each integral in terms of an inverse trigonometric function.<\/p>\n<div id=\"fs-id1170572180188\" class=\"exercise\">\n<div id=\"fs-id1170572180190\" class=\"textbox\">\n<p id=\"fs-id1170572180192\">[latex]{\\int }_{0}^{\\sqrt{3}\\text{\/}2}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572331724\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572331724\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572331724\">[latex]{ \\sin }^{-1}x{|}_{0}^{\\sqrt{3}\\text{\/}2}=\\frac{\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572369231\" class=\"exercise\">\n<div id=\"fs-id1170572369233\" class=\"textbox\">\n<p id=\"fs-id1170572369235\">[latex]{\\int }_{-1\\text{\/}2}^{1\\text{\/}2}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571609291\" class=\"exercise\">\n<div id=\"fs-id1170571660171\" class=\"textbox\">\n<p id=\"fs-id1170571660173\">[latex]{\\int }_{\\sqrt{3}}^{1}\\frac{dx}{\\sqrt{1+{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572331752\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572331752\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572331752\">[latex]{ \\tan }^{-1}x{|}_{\\sqrt{3}}^{1}=-\\frac{\\pi }{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571712219\" class=\"exercise\">\n<div id=\"fs-id1170571712221\" class=\"textbox\">\n<p id=\"fs-id1170571712224\">[latex]{\\int }_{1\\text{\/}\\sqrt{3}}^{\\sqrt{3}}\\frac{dx}{1+{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572625745\" class=\"exercise\">\n<div id=\"fs-id1170572625748\" class=\"textbox\">\n<p id=\"fs-id1170572509844\">[latex]{\\int }_{1}^{\\sqrt{2}}\\frac{dx}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572480541\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572480541\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572480541\">[latex]{ \\sec }^{-1}x{|}_{1}^{\\sqrt{2}}=\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572560653\" class=\"exercise\">\n<div id=\"fs-id1170572560655\" class=\"textbox\">\n<p id=\"fs-id1170572560657\">[latex]{\\int }_{1}^{2\\text{\/}\\sqrt{3}}\\frac{dx}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572333181\">In the following exercises, find each indefinite integral, using appropriate substitutions.<\/p>\n<div id=\"fs-id1170572333184\" class=\"exercise\">\n<div id=\"fs-id1170572333187\" class=\"textbox\">\n<p id=\"fs-id1170572333189\">[latex]\\int \\frac{dx}{\\sqrt{9-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]{ \\sin }^{-1}(\\frac{x}{3})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571543108\" class=\"exercise\">\n<div id=\"fs-id1170572163833\" class=\"textbox\">\n<p id=\"fs-id1170572163835\">[latex]\\int \\frac{dx}{\\sqrt{1-16{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571630356\" class=\"exercise\">\n<div id=\"fs-id1170571630358\" class=\"textbox\">\n<p id=\"fs-id1170571630360\">[latex]\\int \\frac{dx}{9+{x}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572614124\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572614124\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572614124\">[latex]\\frac{1}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{x}{3})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571719603\" class=\"exercise\">\n<div id=\"fs-id1170572425059\" class=\"textbox\">\n<p id=\"fs-id1170572425061\">[latex]\\int \\frac{dx}{25+16{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571712540\" class=\"exercise\">\n<div id=\"fs-id1170571712542\" class=\"textbox\">\n<p id=\"fs-id1170571712545\">[latex]\\int \\frac{dx}{|x|\\sqrt{{x}^{2}-9}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572559663\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572559663\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572559663\">[latex]\\frac{1}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}(\\frac{x}{3})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170571609451\">[latex]\\int \\frac{dx}{|x|\\sqrt{4{x}^{2}-16}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572274931\" class=\"exercise\">\n<div id=\"fs-id1170572274933\" class=\"textbox\">\n<p id=\"fs-id1170572274935\">Explain the relationship [latex]\\text{\u2212}{ \\cos }^{-1}t+C=\\int \\frac{dt}{\\sqrt{1-{t}^{2}}}={ \\sin }^{-1}t+C.[\/latex] Is it true, in general, that [latex]{ \\cos }^{-1}t=\\text{\u2212}{ \\sin }^{-1}t?[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571788133\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571788133\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571788133\">[latex]\\cos (\\frac{\\pi }{2}-\\theta )= \\sin \\theta .[\/latex] So, [latex]{ \\sin }^{-1}t=\\frac{\\pi }{2}-{ \\cos }^{-1}t.[\/latex] They differ by a constant.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572563040\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170572563044\">Explain the relationship [latex]{ \\sec }^{-1}t+C=\\int \\frac{dt}{|t|\\sqrt{{t}^{2}-1}}=\\text{\u2212}{ \\csc }^{-1}t+C.[\/latex] Is it true, in general, that [latex]{ \\sec }^{-1}t=\\text{\u2212}{ \\csc }^{-1}t?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1170571609260\" class=\"textbox\">\n<p id=\"fs-id1170571638317\">Explain what is wrong with the following integral: [latex]{\\int }_{1}^{2}\\frac{dt}{\\sqrt{1-{t}^{2}}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572624827\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572624827\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572624827\">[latex]\\sqrt{1-{t}^{2}}[\/latex] is not defined as a real number when [latex]t>1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572163878\" class=\"exercise\">\n<div id=\"fs-id1170572163880\" class=\"textbox\">\n<p id=\"fs-id1170572163882\">Explain what is wrong with the following integral: [latex]{\\int }_{-1}^{1}\\frac{dt}{|t|\\sqrt{{t}^{2}-1}}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>In the following exercises, solve for the antiderivative [latex]\\int f[\/latex] of [latex]f[\/latex] with [latex]C=0,[\/latex] then use a calculator to graph [latex]f[\/latex] and the antiderivative over the given interval [latex]\\left[a,b\\right].[\/latex] Identify a value of <em>C<\/em> such that adding <em>C<\/em> to the antiderivative recovers the definite integral [latex]F(x)={\\int }_{a}^{x}f(t)dt.[\/latex]<\/p>\n<div id=\"fs-id1170571699012\" class=\"exercise\">\n<div id=\"fs-id1170571699014\" class=\"textbox\">\n<p id=\"fs-id1170571699016\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{\\sqrt{9-{x}^{2}}}dx[\/latex] over [latex]\\left[-3,3\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572274896\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572274896\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1170572274902\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204301\/CNX_Calc_Figure_05_07_201.jpg\" alt=\"Two graphs. The first shows the function f(x) = 1 \/ sqrt(9 \u2013 x^2). It is an upward opening curve symmetric about the y axis, crossing at (0, 1\/3). The second shows the function F(x) = arcsin(1\/3 x). It is an increasing curve going through the origin.\" \/><\/span><br \/>\nThe antiderivative is [latex]{ \\sin }^{-1}(\\frac{x}{3})+C.[\/latex] Taking [latex]C=\\frac{\\pi }{2}[\/latex] recovers the definite integral.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571649965\" class=\"exercise\">\n<div id=\"fs-id1170571649967\" class=\"textbox\">\n<p id=\"fs-id1170571649969\"><strong>[T]<\/strong>[latex]\\int \\frac{9}{9+{x}^{2}}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572565396\" class=\"exercise\">\n<div id=\"fs-id1170572565398\" class=\"textbox\">\n<p><strong>[T]<\/strong>[latex]\\int \\frac{ \\cos x}{4+{ \\sin }^{2}x}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571678881\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571678881\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1170571678884\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204305\/CNX_Calc_Figure_05_07_203.jpg\" alt=\"Two graphs. The first shows the function f(x) = cos(x) \/ (4 + sin(x)^2). It is an oscillating function over &#091;-6, 6&#093; with turning points at roughly (-3, -2.5), (0, .25), and (3, -2.5), where (0,.25) is a local max and the others are local mins. The second shows the function F(x) = .5 * arctan(.5*sin(x)), which also oscillates over &#091;-6,6&#093;. It has turning points at roughly (-4.5, .25), (-1.5, -.25), (1.5, .25), and (4.5, -.25).\" \/><\/span><br \/>\nThe antiderivative is [latex]\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{ \\sin x}{2})+C.[\/latex] Taking [latex]C=\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{ \\sin (6)}{2})[\/latex] recovers the definite integral.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572368670\" class=\"exercise\">\n<div id=\"fs-id1170572368672\" class=\"textbox\">\n<p id=\"fs-id1170572368675\"><strong>[T]<\/strong>[latex]\\int \\frac{{e}^{x}}{1+{e}^{2x}}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170571769590\">In the following exercises, compute the antiderivative using appropriate substitutions.<\/p>\n<div id=\"fs-id1170571769593\" class=\"exercise\">\n<div id=\"fs-id1170571769595\" class=\"textbox\">\n<p id=\"fs-id1170571769597\">[latex]\\int \\frac{{ \\sin }^{-1}tdt}{\\sqrt{1-{t}^{2}}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572611876\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572611876\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572611876\">[latex]\\frac{1}{2}{({ \\sin }^{-1}t)}^{2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571636151\" class=\"exercise\">\n<div id=\"fs-id1170571636153\" class=\"textbox\">\n<p id=\"fs-id1170571636155\">[latex]\\int \\frac{dt}{{ \\sin }^{-1}t\\sqrt{1-{t}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572628413\" class=\"exercise\">\n<div id=\"fs-id1170572628415\" class=\"textbox\">\n<p>[latex]\\int \\frac{{ \\tan }^{-1}(2t)}{1+4{t}^{2}}dt[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1170572628467\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572628467\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572628467\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{1}{4}{({ \\tan }^{-1}(2t))}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572331872\" class=\"exercise\">\n<div id=\"fs-id1170572331874\" class=\"textbox\">\n<p id=\"fs-id1170572331876\">[latex]\\int \\frac{t{ \\tan }^{-1}({t}^{2})}{1+{t}^{4}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572558294\" class=\"exercise\">\n<div id=\"fs-id1170572558296\" class=\"textbox\">\n<p id=\"fs-id1170572558299\">[latex]\\int \\frac{{ \\sec }^{-1}(\\frac{t}{2})}{|t|\\sqrt{{t}^{2}-4}}dt[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572184332\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572184332\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572184332\">[latex]\\frac{1}{4}({ \\sec }^{-1}{(\\frac{t}{2})}^{2})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571733959\" class=\"exercise\">\n<div id=\"fs-id1170571733961\" class=\"textbox\">\n<p id=\"fs-id1170571733963\">[latex]\\int \\frac{t{ \\sec }^{-1}({t}^{2})}{{t}^{2}\\sqrt{{t}^{4}-1}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>In the following exercises, use a calculator to graph the antiderivative [latex]\\int f[\/latex] with [latex]C=0[\/latex] over the given interval [latex]\\left[a,b\\right].[\/latex] Approximate a value of <em>C<\/em>, if possible, such that adding <em>C<\/em> to the antiderivative gives the same value as the definite integral [latex]F(x)={\\int }_{a}^{x}f(t)dt.[\/latex]<\/p>\n<div id=\"fs-id1170572601262\" class=\"exercise\">\n<div id=\"fs-id1170572601264\" class=\"textbox\">\n<p id=\"fs-id1170572601266\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{x\\sqrt{{x}^{2}-4}}dx[\/latex] over [latex]\\left[2,6\\right][\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571679782\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571679782\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1170571679787\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204308\/CNX_Calc_Figure_05_07_205.jpg\" alt=\"A graph of the function f(x) = -.5 * arctan(2 \/ ( sqrt(x^2 \u2013 4) ) ) in quadrant four. It is an increasing concave down curve with a vertical asymptote at x=2.\" \/><\/span><br \/>\nThe antiderivative is [latex]\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\sec }^{-1}(\\frac{x}{2})+C.[\/latex] Taking [latex]C=0[\/latex] recovers the definite integral over [latex]\\left[2,6\\right].[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571679862\" class=\"exercise\">\n<div id=\"fs-id1170571679864\" class=\"textbox\">\n<p id=\"fs-id1170571679866\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{(2x+2)\\sqrt{x}}dx[\/latex] over [latex]\\left[0,6\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572249934\" class=\"exercise\">\n<div id=\"fs-id1170572249937\" class=\"textbox\">\n<p id=\"fs-id1170572249939\"><strong>[T]<\/strong>[latex]\\int \\frac{( \\sin x+x \\cos x)}{1+{x}^{2}{ \\sin }^{2}x}dx[\/latex] over [latex]\\left[-6,6\\right][\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204310\/CNX_Calc_Figure_05_07_207.jpg\" alt=\"The graph of f(x) = arctan(x sin(x)) over [-6,6]. It has five turning points at roughly (-5, -1.5), (-2,1), (0,0), (2,1), and (5,-1.5).\" \/><br \/>\nThe general antiderivative is [latex]{ \\tan }^{-1}(x \\sin x)+C.[\/latex] Taking [latex]C=\\text{\u2212}{ \\tan }^{-1}(6 \\sin (6))[\/latex] recovers the definite integral.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571810858\" class=\"exercise\">\n<div id=\"fs-id1170571810860\" class=\"textbox\">\n<p id=\"fs-id1170571810862\"><strong>[T]<\/strong>[latex]\\int \\frac{2{e}^{-2x}}{\\sqrt{1-{e}^{-4x}}}dx[\/latex] over [latex]\\left[0,2\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571697193\" class=\"exercise\">\n<div id=\"fs-id1170571697196\" class=\"textbox\">\n<p id=\"fs-id1170571697198\"><strong>[T]<\/strong>[latex]\\int \\frac{1}{x+x{\\text{ln}}^{2}x}[\/latex] over [latex]\\left[0,2\\right][\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p><span id=\"fs-id1170571539140\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204313\/CNX_Calc_Figure_05_07_209.jpg\" alt=\"A graph of the function f(x) = arctan(ln(x)) over (0, 2]. It is an increasing curve with x-intercept at (1,0).\" \/><\/span><br \/>\nThe general antiderivative is [latex]{ \\tan }^{-1}(\\text{ln}x)+C.[\/latex] Taking [latex]C=\\frac{\\pi }{2}={ \\tan }^{-1}\\infty[\/latex] recovers the definite integral.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571539209\" class=\"exercise\">\n<div id=\"fs-id1170571543184\" class=\"textbox\">\n<p id=\"fs-id1170571543187\"><strong>[T]<\/strong>[latex]\\int \\frac{{ \\sin }^{-1}x}{\\sqrt{1-{x}^{2}}}[\/latex] over [latex]\\left[-1,1\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572456361\">In the following exercises, compute each integral using appropriate substitutions.<\/p>\n<div id=\"fs-id1170572456364\" class=\"exercise\">\n<div id=\"fs-id1170572456366\" class=\"textbox\">\n<p>[latex]\\int \\frac{{e}^{x}}{\\sqrt{1-{e}^{2t}}}dt[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572456411\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572456411\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572456411\">[latex]{ \\sin }^{-1}({e}^{t})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571580944\" class=\"exercise\">\n<div id=\"fs-id1170571580946\" class=\"textbox\">\n<p>[latex]\\int \\frac{{e}^{t}}{1+{e}^{2t}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572396478\" class=\"exercise\">\n<div id=\"fs-id1170572396480\" class=\"textbox\">\n<p id=\"fs-id1170572396482\">[latex]\\int \\frac{dt}{t\\sqrt{1-{\\text{ln}}^{2}t}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572396521\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572396521\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572396521\">[latex]{ \\sin }^{-1}(\\text{ln}t)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572396552\" class=\"exercise\">\n<div id=\"fs-id1170572396554\" class=\"textbox\">\n<p id=\"fs-id1170572396556\">[latex]\\int \\frac{dt}{t(1+{\\text{ln}}^{2}t)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572218636\" class=\"exercise\">\n<div id=\"fs-id1170572218638\" class=\"textbox\">\n<p id=\"fs-id1170572218640\">[latex]\\int \\frac{{ \\cos }^{-1}(2t)}{\\sqrt{1-4{t}^{2}}}dt[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572386126\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572386126\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572386126\">[latex]-\\frac{1}{2}{({ \\cos }^{-1}(2t))}^{2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572386173\" class=\"exercise\">\n<div id=\"fs-id1170572386175\" class=\"textbox\">\n<p id=\"fs-id1170572386177\">[latex]\\int \\frac{{e}^{t}{ \\cos }^{-1}({e}^{t})}{\\sqrt{1-{e}^{2t}}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572643268\">In the following exercises, compute each definite integral.<\/p>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170572643276\">[latex]{\\int }_{0}^{1\\text{\/}2}\\frac{ \\tan ({ \\sin }^{-1}t)}{\\sqrt{1-{t}^{2}}}dt[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p>[latex]\\frac{1}{2}\\text{ln}(\\frac{4}{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572373740\" class=\"exercise\">\n<div id=\"fs-id1170572373742\" class=\"textbox\">\n<p id=\"fs-id1170572373744\">[latex]{\\int }_{1\\text{\/}4}^{1\\text{\/}2}\\frac{ \\tan ({ \\cos }^{-1}t)}{\\sqrt{1-{t}^{2}}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571710706\" class=\"exercise\">\n<div id=\"fs-id1170571710708\" class=\"textbox\">\n<p>[latex]{\\int }_{0}^{1\\text{\/}2}\\frac{ \\sin ({ \\tan }^{-1}t)}{1+{t}^{2}}dt[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572399013\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572399013\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572399013\">[latex]1-\\frac{2}{\\sqrt{5}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572399032\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1170572399036\">[latex]{\\int }_{0}^{1\\text{\/}2}\\frac{ \\cos ({ \\tan }^{-1}t)}{1+{t}^{2}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1170572415154\" class=\"textbox\">\n<p id=\"fs-id1170572415156\">For [latex]A>0,[\/latex] compute [latex]I(A)={\\int }_{\\text{\u2212}A}^{A}\\frac{dt}{1+{t}^{2}}[\/latex] and evaluate [latex]\\underset{a\\to \\infty }{\\text{lim}}I(A),[\/latex] the area under the graph of [latex]\\frac{1}{1+{t}^{2}}[\/latex] on [latex]\\left[\\text{\u2212}\\infty ,\\infty \\right].[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572168753\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572168753\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572168753\">[latex]2{ \\tan }^{-1}(A)\\to \\pi[\/latex] as [latex]A\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1170572629162\" class=\"textbox\">\n<p id=\"fs-id1170572629164\">For [latex]1<B<\\infty ,[\/latex] compute [latex]I(B)={\\int }_{1}^{B}\\frac{dt}{t\\sqrt{{t}^{2}-1}}[\/latex] and evaluate [latex]\\underset{B\\to \\infty }{\\text{lim}}I(B),[\/latex] the area under the graph of [latex]\\frac{1}{t\\sqrt{{t}^{2}-1}}[\/latex] over [latex]\\left[1,\\infty ).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572569940\" class=\"exercise\">\n<div id=\"fs-id1170572569942\" class=\"textbox\">\n<p id=\"fs-id1170572569944\">Use the substitution [latex]u=\\sqrt{2} \\cot x[\/latex] and the identity [latex]1+{ \\cot }^{2}x={ \\csc }^{2}x[\/latex] to evaluate [latex]\\int \\frac{dx}{1+{ \\cos }^{2}x}.[\/latex] (<em>Hint:<\/em> Multiply the top and bottom of the integrand by [latex]{ \\csc }^{2}x.[\/latex])<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572378997\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572378997\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572378997\">Using the hint, one has [latex]\\int \\frac{{ \\csc }^{2}x}{{ \\csc }^{2}x+{ \\cot }^{2}x}dx=\\int \\frac{{ \\csc }^{2}x}{1+2{ \\cot }^{2}x}dx.[\/latex] Set [latex]u=\\sqrt{2} \\cot x.[\/latex] Then, [latex]du=\\text{\u2212}\\sqrt{2}{ \\csc }^{2}x[\/latex] and the integral is [latex]-\\frac{1}{\\sqrt{2}}\\int \\frac{du}{1+{u}^{2}}=-\\frac{1}{\\sqrt{2}}{ \\tan }^{-1}u+C=\\frac{1}{\\sqrt{2}}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\sqrt{2} \\cot x)+C.[\/latex] If one uses the identity [latex]{ \\tan }^{-1}s+{ \\tan }^{-1}(\\frac{1}{s})=\\frac{\\pi }{2},[\/latex] then this can also be written [latex]\\frac{1}{\\sqrt{2}}\\phantom{\\rule{0.05em}{0ex}}{ \\tan }^{-1}(\\frac{ \\tan x}{\\sqrt{2}})+C.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572310020\" class=\"exercise\">\n<div id=\"fs-id1170572310022\" class=\"textbox\">\n<p id=\"fs-id1170572310024\"><strong>[T]<\/strong> Approximate the points at which the graphs of [latex]f(x)=2{x}^{2}-1[\/latex] and [latex]g(x)={(1+4{x}^{2})}^{-3\\text{\/}2}[\/latex] intersect, and approximate the area between their graphs accurate to three decimal places.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571613771\" class=\"exercise\">\n<div id=\"fs-id1170571613774\" class=\"textbox\">\n<p id=\"fs-id1170571613776\">47. <strong>[T]<\/strong> Approximate the points at which the graphs of [latex]f(x)={x}^{2}-1[\/latex] and [latex]f(x)={x}^{2}-1[\/latex] intersect, and approximate the area between their graphs accurate to three decimal places.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170571661768\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170571661768\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571661768\">[latex]x\\approx \u00b11.13525.[\/latex] The left endpoint estimate with [latex]N=100[\/latex] is 2.796 and these decimals persist for [latex]N=500.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571661806\" class=\"exercise\">\n<div id=\"fs-id1170571661808\" class=\"textbox\">\n<p id=\"fs-id1170571661810\">Use the following graph to prove that [latex]{\\int }_{0}^{x}\\sqrt{1-{t}^{2}}dt=\\frac{1}{2}x\\sqrt{1-{x}^{2}}+\\frac{1}{2}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}x.[\/latex]<\/p>\n<p><span id=\"fs-id1170572331317\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204315\/CNX_Calc_Figure_05_07_211.jpg\" alt=\"A diagram containing two shapes, a wedge from a circle shaded in blue on top of a triangle shaded in brown. The triangle\u2019s hypotenuse is one of the radii edges of the wedge of the circle and is 1 unit long. There is a dotted red line forming a rectangle out of part of the wedge and the triangle, with the hypotenuse of the triangle as the diagonal of the rectangle. The curve of the circle is described by the equation sqrt(1-x^2).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572346977\" class=\"review-exercises\"><\/div>\n","protected":false},"author":311,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1776","chapter","type-chapter","status-publish","hentry"],"part":1684,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1776","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1776\/revisions"}],"predecessor-version":[{"id":2546,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1776\/revisions\/2546"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/1684"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/1776\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=1776"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=1776"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=1776"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=1776"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}