{"id":1580,"date":"2021-07-22T16:20:43","date_gmt":"2021-07-22T16:20:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1580"},"modified":"2022-03-19T04:02:14","modified_gmt":"2022-03-19T04:02:14","slug":"integrating-products-and-powers-of-tanx-and-secx","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/integrating-products-and-powers-of-tanx-and-secx\/","title":{"raw":"Integrating Products and Powers of tanx and secx","rendered":"Integrating Products and Powers of tanx and secx"},"content":{"raw":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve integration problems involving products and powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex]<\/li>\r\n \t<li>Use reduction formulas to solve trigonometric integrals<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section id=\"fs-id1165042484780\" data-depth=\"1\">\r\n<p id=\"fs-id1165042954781\">Before discussing the integration of products and powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex], it is useful to recall the integrals involving [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex] we have already learned:<\/p>\r\n\r\n<ol id=\"fs-id1165043229439\" type=\"1\">\r\n \t<li>[latex]{\\displaystyle\\int}{\\sec}^{2}xdx=\\tan{x}+C[\/latex]<\/li>\r\n \t<li>[latex]{\\displaystyle\\int}\\sec{x}\\tan{x}dx=\\sec{x}+C[\/latex]<\/li>\r\n \t<li>[latex]{\\displaystyle\\int}\\tan{x}dx=\\text{ln}|\\sec{x}|+C[\/latex]<\/li>\r\n \t<li>[latex]{\\displaystyle\\int}\\sec{x}dx=\\text{ln}|\\sec{x}+\\tan{x}|+C[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165042480096\">For most integrals of products and powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex], we rewrite the expression we wish to integrate as the sum or difference of integrals of the form [latex]{\\displaystyle\\int}{\\tan}^{j}x{\\sec}^{2}xdx[\/latex] or [latex]{\\displaystyle\\int}{\\sec}^{j}x\\tan{x}dx[\/latex]. As we see in the following example, we can evaluate these new integrals by using <em data-effect=\"italics\">u<\/em>-substitution.\u00a0 Before doing so, it is useful to note how the Pythagorean Identity implies relationships between other pairs of trigonometric functions.<\/p>\r\n\r\n<div id=\"fs-id1165043183813\" data-type=\"example\">\r\n<div id=\"fs-id1165043183815\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043183817\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox examples\">\r\n<h3>Recall: The Pythagorean Identity<\/h3>\r\n<p id=\"fs-id1170572169681\">For any angle [latex] x [\/latex]:<\/p>\r\n<p style=\"padding-left: 30px;\">[latex] \\sin^2 x + \\cos^2 x = 1 [\/latex]<\/p>\r\nDividing the original equation by [latex] \\cos^2 x [\/latex] and simplifying yields an expression for [latex] \\sec^2 x [\/latex] in terms of [latex] \\tan^2 x [\/latex]:\r\n<p style=\"padding-left: 30px;\">[latex] \\tan^2 x + 1 = \\sec^2 x [\/latex]<\/p>\r\nSubtracting both sides of the equation by [latex] 1 [\/latex] yields an expression for [latex] \\tan^2 x [\/latex] in terms of [latex] \\sec^2 x [\/latex]:\r\n<p style=\"padding-left: 30px;\">[latex] \\tan^2 x = \\sec^2 x - 1 [\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Evaluating [latex]{\\displaystyle\\int}{\\sec}^{j}x\\tan{x}dx[\/latex]<\/h3>\r\n<div id=\"fs-id1165043183817\" data-type=\"problem\">\r\n<p id=\"fs-id1165042735807\">Evaluate [latex]{\\displaystyle\\int}{\\sec}^{5}x\\tan{x}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558499\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558499\"]\r\n<div id=\"fs-id1165042358843\" data-type=\"solution\">\r\n<p id=\"fs-id1165042358846\">Start by rewriting [latex]{\\sec}^{5}x\\tan{x}[\/latex] as [latex]{\\sec}^{4}x\\sec{x}\\tan{x}[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165043249437\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\sec}^{5}x\\tan{x}dx&amp; ={\\displaystyle\\int}{\\sec}^{4}x\\sec{x}\\tan{x}dx\\hfill &amp; &amp; \\text{Let }u=\\sec{x};\\text{then},du=\\sec{x}\\tan{x}dx.\\hfill \\\\ &amp; ={\\displaystyle\\int}{u}^{4}du\\hfill &amp; &amp; \\text{Evaluate the integral}.\\hfill \\\\ &amp; =\\frac{1}{5}{u}^{5}+C\\hfill &amp; &amp; \\text{Substitute }\\sec{x}=u.\\hfill \\\\ &amp; =\\frac{1}{5}{\\sec}^{5}x+C\\hfill &amp; &amp; \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572366188\" class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1170572366192\">You can read some interesting information at <a href=\"https:\/\/en.wikipedia.org\/wiki\/Integral_of_secant_cubed\" target=\"_blank\" rel=\"noopener\">this website to learn about a common integral involving the secant<\/a>.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165042708857\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165042708861\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042708863\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165042708863\" data-type=\"problem\">\r\n<p id=\"fs-id1165042708865\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{5}x{\\sec}^{2}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558299\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558299\"]\r\n<div id=\"fs-id1165042713811\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165042713984\">Let [latex]u=\\tan{x}[\/latex] and [latex]du={\\sec}^{2}x[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558399\"]\r\n<div id=\"fs-id1165042450523\" data-type=\"solution\">\r\n<p id=\"fs-id1165040796357\">[latex]\\frac{1}{6}{\\tan}^{6}x+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165042480062\">We now take a look at the various strategies for integrating products and powers of [latex]\\sec{x}[\/latex] and [latex]\\tan{x}[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165043116405\" class=\"problem-solving\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox examples\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Problem-Solving Strategy: Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex]<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165043373753\">To integrate [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex], use the following strategies:<\/p>\r\n\r\n<ol id=\"fs-id1165042556053\" type=\"1\">\r\n \t<li>If [latex]j[\/latex] is even and [latex]j\\ge 2[\/latex], rewrite [latex]{\\sec}^{j}x={\\sec}^{j - 2}x{\\sec}^{2}x[\/latex] and use [latex]{\\sec}^{2}x={\\tan}^{2}x+1[\/latex] to rewrite [latex]{\\sec}^{j - 2}x[\/latex] in terms of [latex]\\tan{x}[\/latex]. Let [latex]u=\\tan{x}[\/latex] and [latex]du={\\sec}^{2}x[\/latex].<\/li>\r\n \t<li>If [latex]k[\/latex] is odd and [latex]j\\ge 1[\/latex], rewrite [latex]{\\tan}^{k}x{\\sec}^{j}x={\\tan}^{k - 1}x{\\sec}^{j - 1}x\\sec{x}\\tan{x}[\/latex] and use [latex]{\\tan}^{2}x={\\sec}^{2}x - 1[\/latex] to rewrite [latex]{\\tan}^{k - 1}x[\/latex] in terms of [latex]\\sec{x}[\/latex]. Let [latex]u=\\sec{x}[\/latex] and [latex]du=\\sec{x}\\tan{x}dx[\/latex]. (<em data-effect=\"italics\">Note<\/em>: If [latex]j[\/latex] is even and [latex]k[\/latex] is odd, then either strategy 1 or strategy 2 may be used.)<\/li>\r\n \t<li>If [latex]k[\/latex] is odd where [latex]k\\ge 3[\/latex] and [latex]j=0[\/latex], rewrite [latex]{\\tan}^{k}x={\\tan}^{k - 2}x{\\tan}^{2}x={\\tan}^{k - 2}x\\left({\\sec}^{2}x - 1\\right)={\\tan}^{k - 2}x{\\sec}^{2}x-{\\tan}^{k - 2}x[\/latex]. It may be necessary to repeat this process on the [latex]{\\tan}^{k - 2}x[\/latex] term.<\/li>\r\n \t<li>If [latex]k[\/latex] is even and [latex]j[\/latex] is odd, then use [latex]{\\tan}^{2}x={\\sec}^{2}x - 1[\/latex] to express [latex]{\\tan}^{k}x[\/latex] in terms of [latex]\\sec{x}[\/latex]. Use integration by parts to integrate odd powers of [latex]\\sec{x}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042707642\" data-type=\"example\">\r\n<div id=\"fs-id1165042707644\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042707647\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex] when [latex]j[\/latex] is Even<\/h3>\r\n<div id=\"fs-id1165042707642\" data-type=\"example\">\r\n<div id=\"fs-id1165042707644\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042707647\" data-type=\"problem\">\r\n<p id=\"fs-id1165043311928\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{6}x{\\sec}^{4}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558199\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558199\"]\r\n<div id=\"fs-id1165042641608\" data-type=\"solution\">\r\n<p id=\"fs-id1165042641610\">Since the power on [latex]\\sec{x}[\/latex] is even, rewrite [latex]{\\sec}^{4}x={\\sec}^{2}x{\\sec}^{2}x[\/latex] and use [latex]{\\sec}^{2}x={\\tan}^{2}x+1[\/latex] to rewrite the first [latex]{\\sec}^{2}x[\/latex] in terms of [latex]\\tan{x}[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165043423537\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\tan}^{6}x{\\sec}^{4}xdx&amp; ={\\displaystyle\\int}{\\tan}^{6}x\\left({\\tan}^{2}x+1\\right){\\sec}^{2}xdx\\hfill &amp; &amp; \\text{Let }u=\\tan{x}\\text{ and }du={\\sec}^{2}x.\\hfill \\\\ &amp; ={\\displaystyle\\int}{u}^{6}\\left({u}^{2}+1\\right)du\\hfill &amp; &amp; \\text{Expand}.\\hfill \\\\ &amp; ={\\displaystyle\\int}\\left({u}^{8}+{u}^{6}\\right)du\\hfill &amp; &amp; \\text{Evaluate the integral}.\\hfill \\\\ &amp; =\\frac{1}{9}{u}^{9}+\\frac{1}{7}{u}^{7}+C\\hfill &amp; &amp; \\text{Substitute }\\tan{x}=u.\\hfill \\\\ &amp; =\\frac{1}{9}{\\tan}^{9}x+\\frac{1}{7}{\\tan}^{7}x+C.\\hfill &amp; &amp; \\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042527092\" data-type=\"example\">\r\n<div id=\"fs-id1165042527094\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042527096\" data-type=\"problem\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex] when [latex]k[\/latex] is Odd<\/h3>\r\n<div id=\"fs-id1165042527092\" data-type=\"example\">\r\n<div id=\"fs-id1165042527094\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042527096\" data-type=\"problem\">\r\n<p id=\"fs-id1165043272252\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{5}x{\\sec}^{3}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558099\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558099\"]\r\n<div id=\"fs-id1165042709630\" data-type=\"solution\">\r\n<p id=\"fs-id1165042709632\">Since the power on [latex]\\tan{x}[\/latex] is odd, begin by rewriting [latex]{\\tan}^{5}x{\\sec}^{3}x={\\tan}^{4}x{\\sec}^{2}x\\sec{x}\\tan{x}[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165042977525\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccccc}\\hfill {\\tan}^{5}x{\\sec}^{3}x&amp; =\\hfill &amp; {\\tan}^{4}x{\\sec}^{2}x\\sec{x}\\tan{x}.\\hfill &amp; &amp; &amp; \\text{Write }{\\tan}^{4}x={\\left({\\tan}^{2}x\\right)}^{2}.\\hfill \\\\ \\hfill {\\displaystyle\\int}{\\tan}^{5}x{\\sec}^{3}xdx&amp; =\\hfill &amp; {\\displaystyle\\int}{\\left({\\tan}^{2}x\\right)}^{2}{\\sec}^{2}x\\sec{x}\\tan{x}dx\\hfill &amp; &amp; &amp; \\text{Use }{\\tan}^{2}x={\\sec}^{2}x - 1.\\hfill \\\\ &amp; =\\hfill &amp; {\\displaystyle\\int}{\\left({\\sec}^{2}x - 1\\right)}^{2}{\\sec}^{2}x\\sec{x}\\tan{x}dx\\hfill &amp; &amp; &amp; \\text{Let }u=\\sec{x}\\text{and}du=\\sec{x}\\tan{x}dx.\\hfill \\\\ &amp; =\\hfill &amp; {\\displaystyle\\int}{\\left({u}^{2}-1\\right)}^{2}{u}^{2}du\\hfill &amp; &amp; &amp; \\text{Expand}.\\hfill \\\\ &amp; =\\hfill &amp; {\\displaystyle\\int}\\left({u}^{6}-2{u}^{4}+{u}^{2}\\right)du\\hfill &amp; &amp; &amp; \\text{Integrate}.\\hfill \\\\ &amp; =\\hfill &amp; \\frac{1}{7}{u}^{7}-\\frac{2}{5}{u}^{5}+\\frac{1}{3}{u}^{3}+C\\hfill &amp; &amp; &amp; \\text{Substitute }\\sec{x}=u.\\hfill \\\\ &amp; =\\hfill &amp; \\frac{1}{7}{\\sec}^{7}x-\\frac{2}{5}{\\sec}^{5}x+\\frac{1}{3}{\\sec}^{3}x+C.\\hfill &amp; &amp; &amp; \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043327268\" data-type=\"example\">\r\n<div id=\"fs-id1165043327270\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043327272\" data-type=\"problem\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}xdx[\/latex] where [latex]k[\/latex] is Odd and [latex]k\\ge 3[\/latex]<\/h3>\r\n<div id=\"fs-id1165043327272\" data-type=\"problem\">\r\n<p id=\"fs-id1165042638518\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{3}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44557899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44557899\"]\r\n<div id=\"fs-id1165043323904\" data-type=\"solution\">\r\n<p id=\"fs-id1165043323906\">Begin by rewriting [latex]{\\tan}^{3}x=\\tan{x}{\\tan}^{2}x=\\tan{x}\\left({\\sec}^{2}x - 1\\right)=\\tan{x}{\\sec}^{2}x-\\tan{x}[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165042375616\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\tan}^{3}xdx\\hfill &amp; ={\\displaystyle\\int}\\left(\\tan{x}{\\sec}^{2}x-\\tan{x}\\right)dx\\hfill \\\\ \\hfill &amp; ={\\displaystyle\\int}\\tan{x}{\\sec}^{2}xdx-{\\displaystyle\\int}\\tan{x}dx\\hfill \\\\ \\hfill &amp; =\\frac{1}{2}{\\tan}^{2}x-\\text{ln}|\\sec{x}|+C.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043380250\">For the first integral, use the substitution [latex]u=\\tan{x}[\/latex]. For the second integral, use the formula.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043311678\" data-type=\"example\">\r\n<div id=\"fs-id1165043311680\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043311682\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex]<\/h3>\r\n<div id=\"fs-id1165043311682\" data-type=\"problem\">\r\n<p id=\"fs-id1165042509441\">Integrate [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44556899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44556899\"]\r\n<div id=\"fs-id1165042377398\" data-type=\"solution\">\r\n<p id=\"fs-id1165042377401\">This integral requires integration by parts. To begin, let [latex]u=\\sec{x}[\/latex] and [latex]dv={\\sec}^{2}x[\/latex]. These choices make [latex]du=\\sec{x}\\tan{x}[\/latex] and [latex]v=\\tan{x}[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165043161235\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\sec}^{3}xdx&amp; =\\sec{x}\\tan{x}-{\\displaystyle\\int}\\tan{x}\\sec{x}\\tan{x}dx\\hfill &amp; &amp; \\\\ &amp; =\\sec{x}\\tan{x}-{\\displaystyle\\int}{\\tan}^{2}x\\sec{x}dx\\hfill &amp; &amp; \\text{Simplify}.\\hfill \\\\ &amp; =\\sec{x}\\tan{x}-{\\displaystyle\\int}\\left({\\sec}^{2}x - 1\\right)\\sec{x}dx\\hfill &amp; &amp; \\text{Substitute }{\\tan}^{2}x={\\sec}^{2}x - 1.\\hfill \\\\ &amp; =\\sec{x}\\tan{x}+{\\displaystyle\\int}\\sec{x}dx-{\\displaystyle\\int}{\\sec}^{3}xdx\\hfill &amp; &amp; \\text{Rewrite}.\\hfill \\\\ &amp; =\\sec{x}\\tan{x}+\\text{ln}|\\sec{x}+\\tan{x}|-{\\displaystyle\\int}{\\sec}^{3}xdx.\\hfill &amp; &amp; \\text{Evaluate}{\\displaystyle\\int}\\sec{x}dx.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043281629\">We now have<\/p>\r\n\r\n<div id=\"fs-id1165043281632\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\sec}^{3}xdx=\\sec{x}\\tan{x}+\\text{ln}|\\sec{x}+\\tan{x}|-{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042977801\">Since the integral [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex] has reappeared on the right-hand side, we can solve for [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex] by adding it to both sides. In doing so, we obtain<\/p>\r\n\r\n<div id=\"fs-id1165043311020\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]2{\\displaystyle\\int}{\\sec}^{3}xdx=\\sec{x}\\tan{x}+\\text{ln}|\\sec{x}+\\tan{x}|[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043348168\">Dividing by 2, we arrive at<\/p>\r\n\r\n<div id=\"fs-id1165043348172\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\sec}^{3}xdx=\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}\\text{ln}|\\sec{x}+\\tan{x}|+C[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042808795\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165042808799\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042808801\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165042808801\" data-type=\"problem\">\r\n<p id=\"fs-id1165042808804\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{3}x{\\sec}^{7}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44554899\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44554899\"]\r\n<div id=\"fs-id1165042578765\" data-type=\"commentary\" data-element-type=\"hint\">\r\n\r\nUse the previous example as a guide: Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex] when [latex]k[\/latex] is Odd.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44555899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44555899\"]\r\n<div id=\"fs-id1165042611982\" data-type=\"solution\">\r\n<p id=\"fs-id1165040946768\">[latex]\\frac{1}{9}{\\sec}^{9}x-\\frac{1}{7}{\\sec}^{7}x+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OW-JQPR36co?controls=0&amp;start=1957&amp;end=2078&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.2TrigonometricIntegrals1957to2078_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.2 Trigonometric Integrals\" here (opens in new window)<\/a>.\r\n\r\n<\/section><section id=\"fs-id1165042578783\" data-depth=\"1\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]5560[\/ohm_question]\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Reduction Formulas<\/h2>\r\n<p id=\"fs-id1165042578788\">Evaluating [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] for values of [latex]n[\/latex] where [latex]n[\/latex] is odd requires integration by parts. In addition, we must also know the value of [latex]{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex] to evaluate [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex]. The evaluation of [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex] also requires being able to integrate [latex]{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]. To make the process easier, we can derive and apply the following <span data-type=\"term\">power reduction formulas<\/span>. These rules allow us to replace the integral of a power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex] with the integral of a lower power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042511754\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Rule: Reduction Formulas for [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] and [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex]<\/h3>\r\n\r\n<hr \/>\r\n\r\n<div id=\"fs-id1165043276273\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\sec}^{n}xdx=\\frac{1}{n - 1}{\\sec}^{n - 2}x\\tan{x}+\\frac{n - 2}{n - 1}{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex]<\/div>\r\n&nbsp;\r\n<div id=\"fs-id1165039565660\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\tan}^{n}xdx=\\frac{1}{n - 1}{\\tan}^{n - 1}x-{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042445752\">The first power reduction rule may be verified by applying integration by parts. The second may be verified by following the strategy outlined for integrating odd powers of [latex]\\tan{x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042445770\" data-type=\"example\">\r\n<div id=\"fs-id1165042445772\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042445774\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Revisiting [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex]<\/h3>\r\n<div id=\"fs-id1165042445770\" data-type=\"example\">\r\n<div id=\"fs-id1165042445772\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042445774\" data-type=\"problem\">\r\n<p id=\"fs-id1165042977371\">Apply a reduction formula to evaluate [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44553899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44553899\"]\r\n<div id=\"fs-id1165042977399\" data-type=\"solution\">\r\n<p id=\"fs-id1165042977401\">By applying the first reduction formula, we obtain<\/p>\r\n\r\n<div id=\"fs-id1165042977404\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\sec}^{3}xdx\\hfill &amp; =\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}{\\displaystyle\\int}\\sec{x}dx\\hfill \\\\ \\hfill &amp; =\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}\\text{ln}|\\sec{x}+\\tan{x}|+C.\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042599295\" data-type=\"example\">\r\n<div id=\"fs-id1165042599297\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042599300\" data-type=\"problem\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Using a Reduction Formula<\/h3>\r\n<div id=\"fs-id1165042599300\" data-type=\"problem\">\r\n<p id=\"fs-id1165042599305\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{4}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44552899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44552899\"]\r\n<div id=\"fs-id1165042317170\" data-type=\"solution\">\r\n<p id=\"fs-id1165042317172\">Applying the reduction formula for [latex]{\\displaystyle\\int}{\\tan}^{4}xdx[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1165042317197\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\tan}^{4}xdx&amp; =\\frac{1}{3}{\\tan}^{3}x-{\\displaystyle\\int}{\\tan}^{2}xdx\\hfill &amp; &amp; \\\\ &amp; =\\frac{1}{3}{\\tan}^{3}x-\\left(\\tan{x}-{\\displaystyle\\int}{\\tan}^{0}xdx\\right)\\hfill &amp; &amp; \\text{Apply the reduction formula to}{\\displaystyle\\int}{\\tan}^{2}xdx.\\hfill \\\\ &amp; =\\frac{1}{3}{\\tan}^{3}x-\\tan{x}+{\\displaystyle\\int}1dx\\hfill &amp; &amp; \\text{Simplify}.\\hfill \\\\ &amp; =\\frac{1}{3}{\\tan}^{3}x-\\tan{x}+x+C.\\hfill &amp; &amp; \\text{Evaluate}{\\displaystyle\\int}1dx.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043301703\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165043301707\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043301710\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165043301710\" data-type=\"problem\">\r\n<p id=\"fs-id1165043301712\">Apply the reduction formula to [latex]{\\displaystyle\\int}{\\sec}^{5}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44550899\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44550899\"]\r\n<div id=\"fs-id1165042851650\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165042851658\">Use reduction formula 1 and let [latex]n=5[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44551899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44551899\"]\r\n<div id=\"fs-id1165043301739\" data-type=\"solution\">\r\n<p id=\"fs-id1165042259273\">[latex]{\\displaystyle\\int}{\\sec}^{5}xdx=\\frac{1}{4}{\\sec}^{3}x\\tan{x}-\\frac{3}{4}{\\displaystyle\\int}{\\sec}^{3}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OW-JQPR36co?controls=0&amp;start=2244&amp;end=2284&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.2TrigonometricIntegrals2244to2284_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.2 Trigonometric Integrals\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]71966[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve integration problems involving products and powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex]<\/li>\n<li>Use reduction formulas to solve trigonometric integrals<\/li>\n<\/ul>\n<\/div>\n<section id=\"fs-id1165042484780\" data-depth=\"1\">\n<p id=\"fs-id1165042954781\">Before discussing the integration of products and powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex], it is useful to recall the integrals involving [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex] we have already learned:<\/p>\n<ol id=\"fs-id1165043229439\" type=\"1\">\n<li>[latex]{\\displaystyle\\int}{\\sec}^{2}xdx=\\tan{x}+C[\/latex]<\/li>\n<li>[latex]{\\displaystyle\\int}\\sec{x}\\tan{x}dx=\\sec{x}+C[\/latex]<\/li>\n<li>[latex]{\\displaystyle\\int}\\tan{x}dx=\\text{ln}|\\sec{x}|+C[\/latex]<\/li>\n<li>[latex]{\\displaystyle\\int}\\sec{x}dx=\\text{ln}|\\sec{x}+\\tan{x}|+C[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1165042480096\">For most integrals of products and powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex], we rewrite the expression we wish to integrate as the sum or difference of integrals of the form [latex]{\\displaystyle\\int}{\\tan}^{j}x{\\sec}^{2}xdx[\/latex] or [latex]{\\displaystyle\\int}{\\sec}^{j}x\\tan{x}dx[\/latex]. As we see in the following example, we can evaluate these new integrals by using <em data-effect=\"italics\">u<\/em>-substitution.\u00a0 Before doing so, it is useful to note how the Pythagorean Identity implies relationships between other pairs of trigonometric functions.<\/p>\n<div id=\"fs-id1165043183813\" data-type=\"example\">\n<div id=\"fs-id1165043183815\" data-type=\"exercise\">\n<div id=\"fs-id1165043183817\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox examples\">\n<h3>Recall: The Pythagorean Identity<\/h3>\n<p id=\"fs-id1170572169681\">For any angle [latex]x[\/latex]:<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\sin^2 x + \\cos^2 x = 1[\/latex]<\/p>\n<p>Dividing the original equation by [latex]\\cos^2 x[\/latex] and simplifying yields an expression for [latex]\\sec^2 x[\/latex] in terms of [latex]\\tan^2 x[\/latex]:<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\tan^2 x + 1 = \\sec^2 x[\/latex]<\/p>\n<p>Subtracting both sides of the equation by [latex]1[\/latex] yields an expression for [latex]\\tan^2 x[\/latex] in terms of [latex]\\sec^2 x[\/latex]:<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\tan^2 x = \\sec^2 x - 1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Evaluating [latex]{\\displaystyle\\int}{\\sec}^{j}x\\tan{x}dx[\/latex]<\/h3>\n<div id=\"fs-id1165043183817\" data-type=\"problem\">\n<p id=\"fs-id1165042735807\">Evaluate [latex]{\\displaystyle\\int}{\\sec}^{5}x\\tan{x}dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558499\">Show Solution<\/span><\/p>\n<div id=\"q44558499\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042358843\" data-type=\"solution\">\n<p id=\"fs-id1165042358846\">Start by rewriting [latex]{\\sec}^{5}x\\tan{x}[\/latex] as [latex]{\\sec}^{4}x\\sec{x}\\tan{x}[\/latex].<\/p>\n<div id=\"fs-id1165043249437\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\sec}^{5}x\\tan{x}dx& ={\\displaystyle\\int}{\\sec}^{4}x\\sec{x}\\tan{x}dx\\hfill & & \\text{Let }u=\\sec{x};\\text{then},du=\\sec{x}\\tan{x}dx.\\hfill \\\\ & ={\\displaystyle\\int}{u}^{4}du\\hfill & & \\text{Evaluate the integral}.\\hfill \\\\ & =\\frac{1}{5}{u}^{5}+C\\hfill & & \\text{Substitute }\\sec{x}=u.\\hfill \\\\ & =\\frac{1}{5}{\\sec}^{5}x+C\\hfill & & \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572366188\" class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1170572366192\">You can read some interesting information at <a href=\"https:\/\/en.wikipedia.org\/wiki\/Integral_of_secant_cubed\" target=\"_blank\" rel=\"noopener\">this website to learn about a common integral involving the secant<\/a>.<\/p>\n<\/div>\n<div id=\"fs-id1165042708857\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165042708861\" data-type=\"exercise\">\n<div id=\"fs-id1165042708863\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165042708863\" data-type=\"problem\">\n<p id=\"fs-id1165042708865\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{5}x{\\sec}^{2}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558299\">Hint<\/span><\/p>\n<div id=\"q44558299\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042713811\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165042713984\">Let [latex]u=\\tan{x}[\/latex] and [latex]du={\\sec}^{2}x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558399\">Show Solution<\/span><\/p>\n<div id=\"q44558399\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042450523\" data-type=\"solution\">\n<p id=\"fs-id1165040796357\">[latex]\\frac{1}{6}{\\tan}^{6}x+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165042480062\">We now take a look at the various strategies for integrating products and powers of [latex]\\sec{x}[\/latex] and [latex]\\tan{x}[\/latex].<\/p>\n<div id=\"fs-id1165043116405\" class=\"problem-solving\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox examples\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Problem-Solving Strategy: Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex]<\/h3>\n<hr \/>\n<p id=\"fs-id1165043373753\">To integrate [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex], use the following strategies:<\/p>\n<ol id=\"fs-id1165042556053\" type=\"1\">\n<li>If [latex]j[\/latex] is even and [latex]j\\ge 2[\/latex], rewrite [latex]{\\sec}^{j}x={\\sec}^{j - 2}x{\\sec}^{2}x[\/latex] and use [latex]{\\sec}^{2}x={\\tan}^{2}x+1[\/latex] to rewrite [latex]{\\sec}^{j - 2}x[\/latex] in terms of [latex]\\tan{x}[\/latex]. Let [latex]u=\\tan{x}[\/latex] and [latex]du={\\sec}^{2}x[\/latex].<\/li>\n<li>If [latex]k[\/latex] is odd and [latex]j\\ge 1[\/latex], rewrite [latex]{\\tan}^{k}x{\\sec}^{j}x={\\tan}^{k - 1}x{\\sec}^{j - 1}x\\sec{x}\\tan{x}[\/latex] and use [latex]{\\tan}^{2}x={\\sec}^{2}x - 1[\/latex] to rewrite [latex]{\\tan}^{k - 1}x[\/latex] in terms of [latex]\\sec{x}[\/latex]. Let [latex]u=\\sec{x}[\/latex] and [latex]du=\\sec{x}\\tan{x}dx[\/latex]. (<em data-effect=\"italics\">Note<\/em>: If [latex]j[\/latex] is even and [latex]k[\/latex] is odd, then either strategy 1 or strategy 2 may be used.)<\/li>\n<li>If [latex]k[\/latex] is odd where [latex]k\\ge 3[\/latex] and [latex]j=0[\/latex], rewrite [latex]{\\tan}^{k}x={\\tan}^{k - 2}x{\\tan}^{2}x={\\tan}^{k - 2}x\\left({\\sec}^{2}x - 1\\right)={\\tan}^{k - 2}x{\\sec}^{2}x-{\\tan}^{k - 2}x[\/latex]. It may be necessary to repeat this process on the [latex]{\\tan}^{k - 2}x[\/latex] term.<\/li>\n<li>If [latex]k[\/latex] is even and [latex]j[\/latex] is odd, then use [latex]{\\tan}^{2}x={\\sec}^{2}x - 1[\/latex] to express [latex]{\\tan}^{k}x[\/latex] in terms of [latex]\\sec{x}[\/latex]. Use integration by parts to integrate odd powers of [latex]\\sec{x}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707642\" data-type=\"example\">\n<div id=\"fs-id1165042707644\" data-type=\"exercise\">\n<div id=\"fs-id1165042707647\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex] when [latex]j[\/latex] is Even<\/h3>\n<div id=\"fs-id1165042707642\" data-type=\"example\">\n<div id=\"fs-id1165042707644\" data-type=\"exercise\">\n<div id=\"fs-id1165042707647\" data-type=\"problem\">\n<p id=\"fs-id1165043311928\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{6}x{\\sec}^{4}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558199\">Show Solution<\/span><\/p>\n<div id=\"q44558199\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042641608\" data-type=\"solution\">\n<p id=\"fs-id1165042641610\">Since the power on [latex]\\sec{x}[\/latex] is even, rewrite [latex]{\\sec}^{4}x={\\sec}^{2}x{\\sec}^{2}x[\/latex] and use [latex]{\\sec}^{2}x={\\tan}^{2}x+1[\/latex] to rewrite the first [latex]{\\sec}^{2}x[\/latex] in terms of [latex]\\tan{x}[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165043423537\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\tan}^{6}x{\\sec}^{4}xdx& ={\\displaystyle\\int}{\\tan}^{6}x\\left({\\tan}^{2}x+1\\right){\\sec}^{2}xdx\\hfill & & \\text{Let }u=\\tan{x}\\text{ and }du={\\sec}^{2}x.\\hfill \\\\ & ={\\displaystyle\\int}{u}^{6}\\left({u}^{2}+1\\right)du\\hfill & & \\text{Expand}.\\hfill \\\\ & ={\\displaystyle\\int}\\left({u}^{8}+{u}^{6}\\right)du\\hfill & & \\text{Evaluate the integral}.\\hfill \\\\ & =\\frac{1}{9}{u}^{9}+\\frac{1}{7}{u}^{7}+C\\hfill & & \\text{Substitute }\\tan{x}=u.\\hfill \\\\ & =\\frac{1}{9}{\\tan}^{9}x+\\frac{1}{7}{\\tan}^{7}x+C.\\hfill & & \\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042527092\" data-type=\"example\">\n<div id=\"fs-id1165042527094\" data-type=\"exercise\">\n<div id=\"fs-id1165042527096\" data-type=\"problem\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex] when [latex]k[\/latex] is Odd<\/h3>\n<div id=\"fs-id1165042527092\" data-type=\"example\">\n<div id=\"fs-id1165042527094\" data-type=\"exercise\">\n<div id=\"fs-id1165042527096\" data-type=\"problem\">\n<p id=\"fs-id1165043272252\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{5}x{\\sec}^{3}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558099\">Show Solution<\/span><\/p>\n<div id=\"q44558099\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042709630\" data-type=\"solution\">\n<p id=\"fs-id1165042709632\">Since the power on [latex]\\tan{x}[\/latex] is odd, begin by rewriting [latex]{\\tan}^{5}x{\\sec}^{3}x={\\tan}^{4}x{\\sec}^{2}x\\sec{x}\\tan{x}[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165042977525\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccccc}\\hfill {\\tan}^{5}x{\\sec}^{3}x& =\\hfill & {\\tan}^{4}x{\\sec}^{2}x\\sec{x}\\tan{x}.\\hfill & & & \\text{Write }{\\tan}^{4}x={\\left({\\tan}^{2}x\\right)}^{2}.\\hfill \\\\ \\hfill {\\displaystyle\\int}{\\tan}^{5}x{\\sec}^{3}xdx& =\\hfill & {\\displaystyle\\int}{\\left({\\tan}^{2}x\\right)}^{2}{\\sec}^{2}x\\sec{x}\\tan{x}dx\\hfill & & & \\text{Use }{\\tan}^{2}x={\\sec}^{2}x - 1.\\hfill \\\\ & =\\hfill & {\\displaystyle\\int}{\\left({\\sec}^{2}x - 1\\right)}^{2}{\\sec}^{2}x\\sec{x}\\tan{x}dx\\hfill & & & \\text{Let }u=\\sec{x}\\text{and}du=\\sec{x}\\tan{x}dx.\\hfill \\\\ & =\\hfill & {\\displaystyle\\int}{\\left({u}^{2}-1\\right)}^{2}{u}^{2}du\\hfill & & & \\text{Expand}.\\hfill \\\\ & =\\hfill & {\\displaystyle\\int}\\left({u}^{6}-2{u}^{4}+{u}^{2}\\right)du\\hfill & & & \\text{Integrate}.\\hfill \\\\ & =\\hfill & \\frac{1}{7}{u}^{7}-\\frac{2}{5}{u}^{5}+\\frac{1}{3}{u}^{3}+C\\hfill & & & \\text{Substitute }\\sec{x}=u.\\hfill \\\\ & =\\hfill & \\frac{1}{7}{\\sec}^{7}x-\\frac{2}{5}{\\sec}^{5}x+\\frac{1}{3}{\\sec}^{3}x+C.\\hfill & & & \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043327268\" data-type=\"example\">\n<div id=\"fs-id1165043327270\" data-type=\"exercise\">\n<div id=\"fs-id1165043327272\" data-type=\"problem\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}xdx[\/latex] where [latex]k[\/latex] is Odd and [latex]k\\ge 3[\/latex]<\/h3>\n<div id=\"fs-id1165043327272\" data-type=\"problem\">\n<p id=\"fs-id1165042638518\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{3}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44557899\">Show Solution<\/span><\/p>\n<div id=\"q44557899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043323904\" data-type=\"solution\">\n<p id=\"fs-id1165043323906\">Begin by rewriting [latex]{\\tan}^{3}x=\\tan{x}{\\tan}^{2}x=\\tan{x}\\left({\\sec}^{2}x - 1\\right)=\\tan{x}{\\sec}^{2}x-\\tan{x}[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165042375616\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\tan}^{3}xdx\\hfill & ={\\displaystyle\\int}\\left(\\tan{x}{\\sec}^{2}x-\\tan{x}\\right)dx\\hfill \\\\ \\hfill & ={\\displaystyle\\int}\\tan{x}{\\sec}^{2}xdx-{\\displaystyle\\int}\\tan{x}dx\\hfill \\\\ \\hfill & =\\frac{1}{2}{\\tan}^{2}x-\\text{ln}|\\sec{x}|+C.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043380250\">For the first integral, use the substitution [latex]u=\\tan{x}[\/latex]. For the second integral, use the formula.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043311678\" data-type=\"example\">\n<div id=\"fs-id1165043311680\" data-type=\"exercise\">\n<div id=\"fs-id1165043311682\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex]<\/h3>\n<div id=\"fs-id1165043311682\" data-type=\"problem\">\n<p id=\"fs-id1165042509441\">Integrate [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44556899\">Show Solution<\/span><\/p>\n<div id=\"q44556899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042377398\" data-type=\"solution\">\n<p id=\"fs-id1165042377401\">This integral requires integration by parts. To begin, let [latex]u=\\sec{x}[\/latex] and [latex]dv={\\sec}^{2}x[\/latex]. These choices make [latex]du=\\sec{x}\\tan{x}[\/latex] and [latex]v=\\tan{x}[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165043161235\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\sec}^{3}xdx& =\\sec{x}\\tan{x}-{\\displaystyle\\int}\\tan{x}\\sec{x}\\tan{x}dx\\hfill & & \\\\ & =\\sec{x}\\tan{x}-{\\displaystyle\\int}{\\tan}^{2}x\\sec{x}dx\\hfill & & \\text{Simplify}.\\hfill \\\\ & =\\sec{x}\\tan{x}-{\\displaystyle\\int}\\left({\\sec}^{2}x - 1\\right)\\sec{x}dx\\hfill & & \\text{Substitute }{\\tan}^{2}x={\\sec}^{2}x - 1.\\hfill \\\\ & =\\sec{x}\\tan{x}+{\\displaystyle\\int}\\sec{x}dx-{\\displaystyle\\int}{\\sec}^{3}xdx\\hfill & & \\text{Rewrite}.\\hfill \\\\ & =\\sec{x}\\tan{x}+\\text{ln}|\\sec{x}+\\tan{x}|-{\\displaystyle\\int}{\\sec}^{3}xdx.\\hfill & & \\text{Evaluate}{\\displaystyle\\int}\\sec{x}dx.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043281629\">We now have<\/p>\n<div id=\"fs-id1165043281632\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\sec}^{3}xdx=\\sec{x}\\tan{x}+\\text{ln}|\\sec{x}+\\tan{x}|-{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042977801\">Since the integral [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex] has reappeared on the right-hand side, we can solve for [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex] by adding it to both sides. In doing so, we obtain<\/p>\n<div id=\"fs-id1165043311020\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]2{\\displaystyle\\int}{\\sec}^{3}xdx=\\sec{x}\\tan{x}+\\text{ln}|\\sec{x}+\\tan{x}|[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043348168\">Dividing by 2, we arrive at<\/p>\n<div id=\"fs-id1165043348172\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\sec}^{3}xdx=\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}\\text{ln}|\\sec{x}+\\tan{x}|+C[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042808795\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165042808799\" data-type=\"exercise\">\n<div id=\"fs-id1165042808801\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165042808801\" data-type=\"problem\">\n<p id=\"fs-id1165042808804\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{3}x{\\sec}^{7}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44554899\">Hint<\/span><\/p>\n<div id=\"q44554899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042578765\" data-type=\"commentary\" data-element-type=\"hint\">\n<p>Use the previous example as a guide: Integrating [latex]{\\displaystyle\\int}{\\tan}^{k}x{\\sec}^{j}xdx[\/latex] when [latex]k[\/latex] is Odd.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44555899\">Show Solution<\/span><\/p>\n<div id=\"q44555899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042611982\" data-type=\"solution\">\n<p id=\"fs-id1165040946768\">[latex]\\frac{1}{9}{\\sec}^{9}x-\\frac{1}{7}{\\sec}^{7}x+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OW-JQPR36co?controls=0&amp;start=1957&amp;end=2078&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.2TrigonometricIntegrals1957to2078_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.2 Trigonometric Integrals&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<section id=\"fs-id1165042578783\" data-depth=\"1\">\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm5560\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5560&theme=oea&iframe_resize_id=ohm5560&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2 data-type=\"title\">Reduction Formulas<\/h2>\n<p id=\"fs-id1165042578788\">Evaluating [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] for values of [latex]n[\/latex] where [latex]n[\/latex] is odd requires integration by parts. In addition, we must also know the value of [latex]{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex] to evaluate [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex]. The evaluation of [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex] also requires being able to integrate [latex]{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]. To make the process easier, we can derive and apply the following <span data-type=\"term\">power reduction formulas<\/span>. These rules allow us to replace the integral of a power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex] with the integral of a lower power of [latex]\\sec{x}[\/latex] or [latex]\\tan{x}[\/latex].<\/p>\n<div id=\"fs-id1165042511754\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Rule: Reduction Formulas for [latex]{\\displaystyle\\int}{\\sec}^{n}xdx[\/latex] and [latex]{\\displaystyle\\int}{\\tan}^{n}xdx[\/latex]<\/h3>\n<hr \/>\n<div id=\"fs-id1165043276273\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\sec}^{n}xdx=\\frac{1}{n - 1}{\\sec}^{n - 2}x\\tan{x}+\\frac{n - 2}{n - 1}{\\displaystyle\\int}{\\sec}^{n - 2}xdx[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1165039565660\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\displaystyle\\int}{\\tan}^{n}xdx=\\frac{1}{n - 1}{\\tan}^{n - 1}x-{\\displaystyle\\int}{\\tan}^{n - 2}xdx[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042445752\">The first power reduction rule may be verified by applying integration by parts. The second may be verified by following the strategy outlined for integrating odd powers of [latex]\\tan{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042445770\" data-type=\"example\">\n<div id=\"fs-id1165042445772\" data-type=\"exercise\">\n<div id=\"fs-id1165042445774\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Revisiting [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex]<\/h3>\n<div id=\"fs-id1165042445770\" data-type=\"example\">\n<div id=\"fs-id1165042445772\" data-type=\"exercise\">\n<div id=\"fs-id1165042445774\" data-type=\"problem\">\n<p id=\"fs-id1165042977371\">Apply a reduction formula to evaluate [latex]{\\displaystyle\\int}{\\sec}^{3}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44553899\">Show Solution<\/span><\/p>\n<div id=\"q44553899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042977399\" data-type=\"solution\">\n<p id=\"fs-id1165042977401\">By applying the first reduction formula, we obtain<\/p>\n<div id=\"fs-id1165042977404\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\sec}^{3}xdx\\hfill & =\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}{\\displaystyle\\int}\\sec{x}dx\\hfill \\\\ \\hfill & =\\frac{1}{2}\\sec{x}\\tan{x}+\\frac{1}{2}\\text{ln}|\\sec{x}+\\tan{x}|+C.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042599295\" data-type=\"example\">\n<div id=\"fs-id1165042599297\" data-type=\"exercise\">\n<div id=\"fs-id1165042599300\" data-type=\"problem\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Using a Reduction Formula<\/h3>\n<div id=\"fs-id1165042599300\" data-type=\"problem\">\n<p id=\"fs-id1165042599305\">Evaluate [latex]{\\displaystyle\\int}{\\tan}^{4}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44552899\">Show Solution<\/span><\/p>\n<div id=\"q44552899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042317170\" data-type=\"solution\">\n<p id=\"fs-id1165042317172\">Applying the reduction formula for [latex]{\\displaystyle\\int}{\\tan}^{4}xdx[\/latex] we have<\/p>\n<div id=\"fs-id1165042317197\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\tan}^{4}xdx& =\\frac{1}{3}{\\tan}^{3}x-{\\displaystyle\\int}{\\tan}^{2}xdx\\hfill & & \\\\ & =\\frac{1}{3}{\\tan}^{3}x-\\left(\\tan{x}-{\\displaystyle\\int}{\\tan}^{0}xdx\\right)\\hfill & & \\text{Apply the reduction formula to}{\\displaystyle\\int}{\\tan}^{2}xdx.\\hfill \\\\ & =\\frac{1}{3}{\\tan}^{3}x-\\tan{x}+{\\displaystyle\\int}1dx\\hfill & & \\text{Simplify}.\\hfill \\\\ & =\\frac{1}{3}{\\tan}^{3}x-\\tan{x}+x+C.\\hfill & & \\text{Evaluate}{\\displaystyle\\int}1dx.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043301703\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165043301707\" data-type=\"exercise\">\n<div id=\"fs-id1165043301710\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165043301710\" data-type=\"problem\">\n<p id=\"fs-id1165043301712\">Apply the reduction formula to [latex]{\\displaystyle\\int}{\\sec}^{5}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44550899\">Hint<\/span><\/p>\n<div id=\"q44550899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042851650\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165042851658\">Use reduction formula 1 and let [latex]n=5[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44551899\">Show Solution<\/span><\/p>\n<div id=\"q44551899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043301739\" data-type=\"solution\">\n<p id=\"fs-id1165042259273\">[latex]{\\displaystyle\\int}{\\sec}^{5}xdx=\\frac{1}{4}{\\sec}^{3}x\\tan{x}-\\frac{3}{4}{\\displaystyle\\int}{\\sec}^{3}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OW-JQPR36co?controls=0&amp;start=2244&amp;end=2284&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.2TrigonometricIntegrals2244to2284_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.2 Trigonometric Integrals&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm71966\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=71966&theme=oea&iframe_resize_id=ohm71966&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1580\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>3.2 Trigonometric Integrals. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"3.2 Trigonometric Integrals\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1580","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1580","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":10,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1580\/revisions"}],"predecessor-version":[{"id":2063,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1580\/revisions\/2063"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/158"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1580\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1580"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1580"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1580"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}