{"id":805,"date":"2021-06-02T23:32:57","date_gmt":"2021-06-02T23:32:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=805"},"modified":"2025-02-25T18:26:08","modified_gmt":"2025-02-25T18:26:08","slug":"integrating-products-and-powers-of-sinx-and-cosx","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/integrating-products-and-powers-of-sinx-and-cosx\/","title":{"raw":"Integrating Products and Powers of sinx and cosx","rendered":"Integrating Products and Powers of sinx and cosx"},"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]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section id=\"fs-id1165042818414\" data-depth=\"1\">\r\n<p id=\"fs-id1165042469843\">A key idea behind the strategy used to integrate combinations of products and powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex] involves rewriting these expressions as sums and differences of integrals of the form [latex]\\displaystyle\\int\\sin^{j}x\\cos{x}dx[\/latex] or [latex]{\\displaystyle\\int}{\\cos}^{j}x\\sin{x}dx[\/latex]. After rewriting these integrals, we evaluate them using <em data-effect=\"italics\">u<\/em>-substitution.<\/p>\r\nBefore describing the general process in detail, let\u2019s take a look at the following examples.\r\n<div id=\"fs-id1165043250980\" data-type=\"example\">\r\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\cos}^{j}x\\sin{x}dx[\/latex]<\/h3>\r\n<div id=\"fs-id1165043250980\" data-type=\"example\">\r\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\r\n<p id=\"fs-id1165042320064\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{3}x\\sin{x}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558899\"]\r\n<div id=\"fs-id1165043081174\" data-type=\"solution\">\r\n<p id=\"fs-id1165042358050\">Use [latex]u[\/latex] -substitution and let [latex]u=\\cos{x}[\/latex]. In this case, [latex]du=\\text{-}\\sin{x}dx[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165043093877\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\cos}^{3}x\\sin{x}dx\\hfill &amp; =\\text{-}{\\displaystyle\\int}{u}^{3}du\\hfill \\\\ \\hfill &amp; =-\\frac{1}{4}{u}^{4}+C\\hfill \\\\ \\hfill &amp; =-\\frac{1}{4}{\\cos}^{4}x+C.\\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042964834\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165043354058\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042904777\" data-type=\"problem\">\r\n<p id=\"fs-id1165043112993\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165043250980\" data-type=\"example\">\r\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n\r\n<span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{4}x\\cos{x}dx[\/latex].<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042964834\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165043354058\" data-type=\"exercise\">\r\n\r\n[reveal-answer q=\"44558897\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1165043428113\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1165042364488\">Let [latex]u=\\sin{x}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1165043051946\" data-type=\"solution\">\r\n<p id=\"fs-id1165040774446\" style=\"text-align: center;\">[latex]\\frac{1}{5}{\\sin}^{5}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\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=118&amp;end=159&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\/3.2TrigonometricIntegrals118to159_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.2 Trigonometric Integrals\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1165043281403\" data-type=\"example\">\r\n<div id=\"fs-id1165043257290\" data-type=\"exercise\">\r\n<div data-type=\"title\">In addition to the technique of [latex]u-[\/latex] substitution, the problems in this section and the next make frequent use of the Pythagorean Identity and its implications for how to rewrite trigonometric functions in terms of other trigonometric functions. We briefly review the relationships between these functions below.<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\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[latex] \\sin^2 x + \\cos^2 x = 1 [\/latex]\r\n\r\nSubtracting by [latex] \\sin^2 x [\/latex] allows a square power of cosine in terms of sine:\r\n[latex] \\cos^2 x = 1-\\sin^2 x [\/latex]\r\n\r\nSubtracting instead by [latex] \\cos^2 x [\/latex] allows a square power of sine to be written in terms of cosine:\r\n\r\n[latex] \\sin^2 x = 1 -\\cos^2 x [\/latex]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165043281403\" data-type=\"example\">\r\n<div id=\"fs-id1165043257290\" data-type=\"exercise\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:<span style=\"font-size: 1em;\"> Integrating [latex]{\\displaystyle\\int}{\\cos}^{j}x{\\sin}^{k}xdx[\/latex] Where <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">k<\/em><span style=\"font-size: 1em;\"> is Odd<\/span><\/h3>\r\n<div id=\"fs-id1165043250980\" data-type=\"example\">\r\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\r\n<div data-type=\"title\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043281403\" data-type=\"example\">\r\n<div id=\"fs-id1165043257290\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043078643\" data-type=\"problem\">\r\n<p id=\"fs-id1165042873576\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{2}x{\\sin}^{3}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558896\"]\r\n<div id=\"fs-id1165042621331\" data-type=\"solution\">\r\n<p id=\"fs-id1165043035592\">To convert this integral to integrals of the form [latex]{\\displaystyle\\int}{\\cos}^{j}x\\sin{x}dx[\/latex], rewrite [latex]{\\sin}^{3}x={\\sin}^{2}x\\sin{x}[\/latex] and make the substitution [latex]{\\sin}^{2}x=1-{\\cos}^{2}x[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165042880012\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}{\\displaystyle\\int}{\\cos}^{2}x{\\sin}^{3}xdx\\hfill &amp; ={\\displaystyle\\int}{\\cos}^{2}x\\left(1-{\\cos}^{2}x\\right)\\sin{x}dx\\hfill &amp; \\text{Let }u=\\cos{x};\\text{then }du=\\text{-}\\sin{x}dx.\\hfill \\\\ \\hfill &amp; =\\text{-}{\\displaystyle\\int}{u}^{2}\\left(1-{u}^{2}\\right)du\\hfill &amp; \\hfill \\\\ \\hfill &amp; ={\\displaystyle\\int}\\left({u}^{4}-{u}^{2}\\right)du\\hfill &amp; \\hfill \\\\ \\hfill &amp; =\\frac{1}{5}{u}^{5}-\\frac{1}{3}{u}^{3}+C\\hfill &amp; \\hfill \\\\ \\hfill &amp; =\\frac{1}{5}{\\cos}^{5}x-\\frac{1}{3}{\\cos}^{3}x+C.\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165043250980\" data-type=\"example\">\r\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{3}x{\\sin}^{2}xdx[\/latex].<\/span>\r\n\r\n[reveal-answer q=\"44558894\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1165043085127\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1165042966824\" style=\"text-align: left;\">Write [latex]{\\cos}^{3}x={\\cos}^{2}x\\cos{x}=\\left(1-{\\sin}^{2}x\\right)\\cos{x}[\/latex] and let [latex]u=\\sin{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<div id=\"fs-id1165043248784\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165042980470\" data-type=\"exercise\">[reveal-answer q=\"44558895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1165043103949\" data-type=\"solution\">\r\n<p id=\"fs-id1165042190535\" style=\"text-align: center;\">[latex]\\frac{1}{3}{\\sin}^{3}x-\\frac{1}{5}{\\sin}^{5}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\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OW-JQPR36co?controls=0&amp;start=337&amp;end=450&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\/3.2TrigonometricIntegrals337to450_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<span style=\"font-size: 1rem; text-align: initial;\">In the next example, we see the strategy that must be applied when there are only even powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex]. For integrals of this type, the identities<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043066788\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)=\\frac{1-\\cos\\left(2x\\right)}{2}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042530463\">and<\/p>\r\n\r\n<div id=\"fs-id1165043331511\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\cos}^{2}x=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(2x\\right)=\\frac{1+\\cos\\left(2x\\right)}{2}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042368394\">are invaluable. These identities are sometimes known as <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">power-reducing identities<\/em><\/span> and they may be derived from the double-angle identity [latex]\\cos\\left(2x\\right)={\\cos}^{2}x-{\\sin}^{2}x[\/latex] and the Pythagorean identity [latex]{\\cos}^{2}x+{\\sin}^{2}x=1[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042936506\" data-type=\"example\">\r\n<div id=\"fs-id1165042318645\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042632772\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>example:\u00a0Integrating an Even Power of [latex]\\sin{x}[\/latex]<\/h3>\r\n<div id=\"fs-id1165042632772\" data-type=\"problem\">\r\n<p id=\"fs-id1165042647060\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{2}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558892\"]\r\n<div id=\"fs-id1165042965476\" data-type=\"solution\">\r\n<p id=\"fs-id1165043057271\">To evaluate this integral, let\u2019s use the trigonometric identity [latex]{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165043194450\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\sin}^{2}xdx\\hfill &amp; ={\\displaystyle\\int}\\left(\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)\\right)dx\\hfill \\\\ \\hfill &amp; =\\frac{1}{2}x-\\frac{1}{4}\\sin\\left(2x\\right)+C.\\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165042936506\" data-type=\"example\">\r\n<div id=\"fs-id1165042318645\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042632772\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">Evaluate [latex]\\displaystyle\\int {\\cos}^{2}xdx[\/latex].<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043096050\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165042989371\" data-type=\"exercise\">\r\n\r\n[reveal-answer q=\"44558890\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558890\"]\r\n<div id=\"fs-id1165043272814\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<div id=\"fs-id1165042375699\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{\\cos}^{2}x=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(2x\\right)[\/latex]<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558891\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558891\"]\r\n<div id=\"fs-id1165042982086\" data-type=\"solution\">\r\n<p id=\"fs-id1165042137283\">[latex]\\frac{1}{2}x+\\frac{1}{4}\\sin\\left(2x\\right)+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 class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]25554[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"font-size: 1rem; text-align: initial;\">The general process for integrating products of powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex] is summarized in the following set of guidelines.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042604945\" 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 Products and Powers of sin <em data-effect=\"italics\">x<\/em> and cos <em data-effect=\"italics\">x<\/em><\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165042275182\">To integrate [latex]{\\displaystyle\\int}{\\cos}^{j}x{\\sin}^{k}xdx[\/latex] use the following strategies:<\/p>\r\n\r\n<ol id=\"fs-id1165043034337\" type=\"1\">\r\n \t<li>If [latex]k[\/latex] is odd, rewrite [latex]{\\sin}^{k}x={\\sin}^{k - 1}x\\sin{x}[\/latex] and use the identity [latex]{\\sin}^{2}x=1-{\\cos}^{2}x[\/latex] to rewrite [latex]{\\sin}^{k - 1}x[\/latex] in terms of [latex]\\cos{x}[\/latex]. Integrate using the substitution [latex]u=\\cos{x}[\/latex]. This substitution makes [latex]du=\\text{-}\\sin{x}dx[\/latex].<\/li>\r\n \t<li>If [latex]j[\/latex] is odd, rewrite [latex]{\\cos}^{j}x={\\cos}^{j - 1}x\\cos{x}[\/latex] and use the identity [latex]{\\cos}^{2}x=1-{\\sin}^{2}x[\/latex] to rewrite [latex]{\\cos}^{j - 1}x[\/latex] in terms of [latex]\\sin{x}[\/latex]. Integrate using the substitution [latex]u=\\sin{x}[\/latex]. This substitution makes [latex]du=\\cos{x}dx[\/latex]. (<em data-effect=\"italics\">Note<\/em>: If both [latex]j[\/latex] and [latex]k[\/latex] are odd, either strategy 1 or strategy 2 may be used.)<\/li>\r\n \t<li>If both [latex]j[\/latex] and [latex]k[\/latex] are even, use [latex]{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)[\/latex] and [latex]{\\cos}^{2}x=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(2x\\right)[\/latex]. After applying these formulas, simplify and reapply strategies 1 through 3 as appropriate.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043272282\" data-type=\"example\">\r\n<div id=\"fs-id1165043086427\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043393170\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Integrating [latex]\\displaystyle\\int {\\cos}^{j}x{\\sin}^{k}xdx[\/latex] where <em data-effect=\"italics\">k<\/em> is Odd<\/h3>\r\n<div id=\"fs-id1165043393170\" data-type=\"problem\">\r\n<p id=\"fs-id1165042907373\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{8}x{\\sin}^{5}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558889\"]\r\n<div id=\"fs-id1165043350731\" data-type=\"solution\">\r\n<p id=\"fs-id1165042505411\">Since the power on [latex]\\sin{x}[\/latex] is odd, use strategy 1. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165042349814\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\cos}^{8}x{\\sin}^{5}xdx&amp; ={\\displaystyle\\int}{\\cos}^{8}x{\\sin}^{4}x\\sin{x}dx\\hfill &amp; &amp; \\text{Break off }\\sin{x}.\\hfill \\\\ &amp; ={\\displaystyle\\int}{\\cos}^{8}x{\\left({\\sin}^{2}x\\right)}^{2}\\sin{x}dx\\hfill &amp; &amp; \\text{Rewrite }{\\sin}^{4}x={\\left({\\sin}^{2}x\\right)}^{2}.\\hfill \\\\ &amp; ={\\displaystyle\\int}{\\cos}^{8}x{\\left(1-{\\cos}^{2}x\\right)}^{2}\\sin{x}dx\\hfill &amp; &amp; \\text{Substitute }{\\sin}^{2}x=1-{\\cos}^{2}x.\\hfill \\\\ &amp; ={\\displaystyle\\int}{u}^{8}{\\left(1-{u}^{2}\\right)}^{2}\\left(\\text{-}du\\right)\\hfill &amp; &amp; \\text{Let }u=\\cos{x}\\text{ and }du=\\text{-}\\sin{x}dx.\\hfill \\\\ &amp; ={\\displaystyle\\int}\\left(\\text{-}{u}^{8}+2{u}^{10}-{u}^{12}\\right)du\\hfill &amp; &amp; \\text{Expand}.\\hfill \\\\ &amp; =-\\frac{1}{9}{u}^{9}+\\frac{2}{11}{u}^{11}-\\frac{1}{13}{u}^{13}+C\\hfill &amp; &amp; \\text{Evaluate the integral}.\\hfill \\\\ &amp; =-\\frac{1}{9}{\\cos}^{9}x+\\frac{2}{11}{\\cos}^{11}x-\\frac{1}{13}{\\cos}^{13}x+C.\\hfill &amp; &amp; \\text{Substitute }u=\\cos{x}.\\hfill \\end{array}[\/latex]<\/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-id1165042707196\" data-type=\"example\">\r\n<div id=\"fs-id1165042707198\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042329519\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\cos}^{j}x{\\sin}^{k}xdx[\/latex] where <em data-effect=\"italics\">k<\/em> and <em data-effect=\"italics\">j<\/em> are Even<\/h3>\r\n<div id=\"fs-id1165042329519\" data-type=\"problem\">\r\n<p id=\"fs-id1165043179875\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{4}xdx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558879\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558879\"]\r\n<div id=\"fs-id1165042832305\" data-type=\"solution\">\r\n<p id=\"fs-id1165042832307\">Since the power on [latex]\\sin{x}[\/latex] is even [latex]\\left(k=4\\right)[\/latex] and the power on [latex]\\cos{x}[\/latex] is even [latex]\\left(j=0\\right)[\/latex], we must use strategy 3. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165043089888\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\sin}^{4}xdx&amp; ={\\displaystyle\\int}{\\left({\\sin}^{2}x\\right)}^{2}dx\\hfill &amp; &amp; \\text{Rewrite }{\\sin}^{4}x={\\left({\\sin}^{2}x\\right)}^{2}.\\hfill \\\\ &amp; ={\\displaystyle\\int}{\\left(\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)\\right)}^{2}dx\\hfill &amp; &amp; \\text{Substitute }{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right).\\hfill \\\\ &amp; ={\\displaystyle\\int}\\left(\\frac{1}{4}-\\frac{1}{2}\\cos\\left(2x\\right)+\\frac{1}{4}{\\cos}^{2}\\left(2x\\right)\\right)dx\\hfill &amp; &amp; \\text{Expand }{\\left(\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)\\right)}^{2}.\\hfill \\\\ &amp; ={\\displaystyle\\int}\\left(\\frac{1}{4}-\\frac{1}{2}\\cos\\left(2x\\right)+\\frac{1}{4}\\left(\\frac{1}{2}+\\frac{1}{2}\\cos\\left(4x\\right)\\right)\\right)dx.\\hfill &amp; &amp; \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043074297\">Since [latex]{\\cos}^{2}\\left(2x\\right)[\/latex] has an even power, substitute [latex]{\\cos}^{2}\\left(2x\\right)=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(4x\\right)\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165042508466\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}={\\displaystyle\\int}\\left(\\frac{3}{8}-\\frac{1}{2}\\cos\\left(2x\\right)+\\frac{1}{8}\\cos\\left(4x\\right)\\right)dx\\hfill &amp; \\text{Simplify}.\\hfill \\\\ =\\frac{3}{8}x-\\frac{1}{4}\\sin\\left(2x\\right)+\\frac{1}{32}\\sin\\left(4x\\right)+C\\hfill &amp; \\text{Evaluate the integral}.\\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165042707196\" data-type=\"example\">\r\n<div id=\"fs-id1165042707198\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042329519\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{3}xdx[\/latex].<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043210017\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165042369121\" data-type=\"exercise\">\r\n\r\n[reveal-answer q=\"44558859\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558859\"]\r\n<div id=\"fs-id1165043174075\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165042617688\">Use strategy 2. Write [latex]{\\cos}^{3}x={\\cos}^{2}x\\cos{x}[\/latex] and substitute [latex]{\\cos}^{2}x=1-{\\sin}^{2}x[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558869\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558869\"]\r\n<div id=\"fs-id1165043113739\" data-type=\"solution\">\r\n<p id=\"fs-id1165041795566\">[latex]\\sin{x}-\\frac{1}{3}{\\sin}^{3}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<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165039562343\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165039562346\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042518537\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165042518537\" data-type=\"problem\">\r\n<p id=\"fs-id1165042518539\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{2}\\left(3x\\right)dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558839\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558839\"]\r\n<div id=\"fs-id1165042677393\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165043254305\">Use strategy 3. Substitute [latex]{\\cos}^{2}\\left(3x\\right)=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(6x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558849\"]\r\n<div id=\"fs-id1165042834041\" data-type=\"solution\">\r\n<p id=\"fs-id1165040796562\">[latex]\\frac{1}{2}x+\\frac{1}{12}\\sin\\left(6x\\right)+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-id1165043131551\">In some areas of physics, such as quantum mechanics, signal processing, and the computation of Fourier series, it is often necessary to integrate products that include [latex]\\sin\\left(ax\\right)[\/latex], [latex]\\sin\\left(bx\\right)[\/latex], [latex]\\cos\\left(ax\\right)[\/latex], and [latex]\\cos\\left(bx\\right)[\/latex]. These integrals are evaluated by applying trigonometric identities, as outlined in the following rule.<\/p>\r\n\r\n<div id=\"fs-id1165042832676\" 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\"><strong>Rule: Integrating Products of Sines and Cosines of Different Angles<\/strong><\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165043248760\">To integrate products involving [latex]\\sin\\left(ax\\right)[\/latex], [latex]\\sin\\left(bx\\right)[\/latex], [latex]\\cos\\left(ax\\right)[\/latex], and [latex]\\cos\\left(bx\\right)[\/latex], use the substitutions<\/p>\r\n\r\n<div id=\"fs-id1165043229277\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\sin\\left(ax\\right)\\sin\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)-\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/div>\r\n<div id=\"fs-id1165042349512\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\sin\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\sin\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\sin\\left(\\left(a+b\\right)x\\right)[\/latex]<\/div>\r\n<div id=\"fs-id1165042332192\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\cos\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/div>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165043094088\">These formulas may be derived from the sum-of-angle formulas for sine and cosine.<\/p>\r\n\r\n<div id=\"fs-id1165043094091\" data-type=\"example\">\r\n<div id=\"fs-id1165043321357\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043321359\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Evaluating [latex]\\displaystyle\\int \\sin\\left(ax\\right)\\cos\\left(bx\\right)dx[\/latex]<\/h3>\r\n<div id=\"fs-id1165043321359\" data-type=\"problem\">\r\n<p id=\"fs-id1165043272050\">Evaluate [latex]{\\displaystyle\\int}\\sin\\left(5x\\right)\\cos\\left(3x\\right)dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558799\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558799\"]\r\n<div id=\"fs-id1165043181525\" data-type=\"solution\">\r\n<p id=\"fs-id1165043181527\">Apply the identity [latex]\\sin\\left(5x\\right)\\cos\\left(3x\\right)=\\frac{1}{2}\\sin\\left(2x\\right)+\\frac{1}{2}\\sin\\left(8x\\right)[\/latex]. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165042349275\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}\\sin\\left(5x\\right)\\cos\\left(3x\\right)dx\\hfill &amp; ={\\displaystyle\\int}\\frac{1}{2}\\sin\\left(2x\\right)+\\frac{1}{2}\\sin\\left(8x\\right)dx\\hfill \\\\ \\hfill &amp; =-\\frac{1}{4}\\cos\\left(2x\\right)-\\frac{1}{16}\\cos\\left(8x\\right)+C.\\hfill \\end{array}[\/latex]<\/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-id1165042707106\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165043094213\" data-type=\"exercise\">\r\n<div id=\"fs-id1165043094215\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1165043094215\" data-type=\"problem\">\r\n<p id=\"fs-id1165043094217\">Evaluate [latex]{\\displaystyle\\int}\\cos\\left(6x\\right)\\cos\\left(5x\\right)dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558599\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558599\"]\r\n<div id=\"fs-id1165042644355\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165042644361\">Substitute [latex]\\cos\\left(6x\\right)\\cos\\left(5x\\right)=\\frac{1}{2}\\cos{x}+\\frac{1}{2}\\cos\\left(11x\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558699\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558699\"]\r\n<div id=\"fs-id1165043395284\" data-type=\"solution\">\r\n<p id=\"fs-id1165040670224\">[latex]\\frac{1}{2}\\sin{x}+\\frac{1}{22}\\sin\\left(11x\\right)+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<\/div>\r\n<\/div>\r\n<\/section>","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]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section id=\"fs-id1165042818414\" data-depth=\"1\">\n<p id=\"fs-id1165042469843\">A key idea behind the strategy used to integrate combinations of products and powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex] involves rewriting these expressions as sums and differences of integrals of the form [latex]\\displaystyle\\int\\sin^{j}x\\cos{x}dx[\/latex] or [latex]{\\displaystyle\\int}{\\cos}^{j}x\\sin{x}dx[\/latex]. After rewriting these integrals, we evaluate them using <em data-effect=\"italics\">u<\/em>-substitution.<\/p>\n<p>Before describing the general process in detail, let\u2019s take a look at the following examples.<\/p>\n<div id=\"fs-id1165043250980\" data-type=\"example\">\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\cos}^{j}x\\sin{x}dx[\/latex]<\/h3>\n<div id=\"fs-id1165043250980\" data-type=\"example\">\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\n<p id=\"fs-id1165042320064\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{3}x\\sin{x}dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558899\">Show Solution<\/span><\/p>\n<div id=\"q44558899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043081174\" data-type=\"solution\">\n<p id=\"fs-id1165042358050\">Use [latex]u[\/latex] -substitution and let [latex]u=\\cos{x}[\/latex]. In this case, [latex]du=\\text{-}\\sin{x}dx[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165043093877\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\cos}^{3}x\\sin{x}dx\\hfill & =\\text{-}{\\displaystyle\\int}{u}^{3}du\\hfill \\\\ \\hfill & =-\\frac{1}{4}{u}^{4}+C\\hfill \\\\ \\hfill & =-\\frac{1}{4}{\\cos}^{4}x+C.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042964834\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165043354058\" data-type=\"exercise\">\n<div id=\"fs-id1165042904777\" data-type=\"problem\">\n<p id=\"fs-id1165043112993\">\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165043250980\" data-type=\"example\">\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\n<div data-type=\"title\">\n<p><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{4}x\\cos{x}dx[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042964834\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165043354058\" data-type=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558897\">Hint<\/span><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043428113\" data-type=\"commentary\" data-element-type=\"hint\">\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1165042364488\">Let [latex]u=\\sin{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=\"q44558898\">Show Solution<\/span><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043051946\" data-type=\"solution\">\n<p id=\"fs-id1165040774446\" style=\"text-align: center;\">[latex]\\frac{1}{5}{\\sin}^{5}x+C[\/latex]<\/p>\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=118&amp;end=159&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\/3.2TrigonometricIntegrals118to159_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<div id=\"fs-id1165043281403\" data-type=\"example\">\n<div id=\"fs-id1165043257290\" data-type=\"exercise\">\n<div data-type=\"title\">In addition to the technique of [latex]u-[\/latex] substitution, the problems in this section and the next make frequent use of the Pythagorean Identity and its implications for how to rewrite trigonometric functions in terms of other trigonometric functions. We briefly review the relationships between these functions below.<\/div>\n<\/div>\n<\/div>\n<\/div>\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>[latex]\\sin^2 x + \\cos^2 x = 1[\/latex]<\/p>\n<p>Subtracting by [latex]\\sin^2 x[\/latex] allows a square power of cosine in terms of sine:<br \/>\n[latex]\\cos^2 x = 1-\\sin^2 x[\/latex]<\/p>\n<p>Subtracting instead by [latex]\\cos^2 x[\/latex] allows a square power of sine to be written in terms of cosine:<\/p>\n<p>[latex]\\sin^2 x = 1 -\\cos^2 x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043281403\" data-type=\"example\">\n<div id=\"fs-id1165043257290\" data-type=\"exercise\">\n<div class=\"textbox exercises\">\n<h3>Example:<span style=\"font-size: 1em;\"> Integrating [latex]{\\displaystyle\\int}{\\cos}^{j}x{\\sin}^{k}xdx[\/latex] Where <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">k<\/em><span style=\"font-size: 1em;\"> is Odd<\/span><\/h3>\n<div id=\"fs-id1165043250980\" data-type=\"example\">\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\n<div data-type=\"title\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043281403\" data-type=\"example\">\n<div id=\"fs-id1165043257290\" data-type=\"exercise\">\n<div id=\"fs-id1165043078643\" data-type=\"problem\">\n<p id=\"fs-id1165042873576\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{2}x{\\sin}^{3}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558896\">Show Solution<\/span><\/p>\n<div id=\"q44558896\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042621331\" data-type=\"solution\">\n<p id=\"fs-id1165043035592\">To convert this integral to integrals of the form [latex]{\\displaystyle\\int}{\\cos}^{j}x\\sin{x}dx[\/latex], rewrite [latex]{\\sin}^{3}x={\\sin}^{2}x\\sin{x}[\/latex] and make the substitution [latex]{\\sin}^{2}x=1-{\\cos}^{2}x[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165042880012\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}{\\displaystyle\\int}{\\cos}^{2}x{\\sin}^{3}xdx\\hfill & ={\\displaystyle\\int}{\\cos}^{2}x\\left(1-{\\cos}^{2}x\\right)\\sin{x}dx\\hfill & \\text{Let }u=\\cos{x};\\text{then }du=\\text{-}\\sin{x}dx.\\hfill \\\\ \\hfill & =\\text{-}{\\displaystyle\\int}{u}^{2}\\left(1-{u}^{2}\\right)du\\hfill & \\hfill \\\\ \\hfill & ={\\displaystyle\\int}\\left({u}^{4}-{u}^{2}\\right)du\\hfill & \\hfill \\\\ \\hfill & =\\frac{1}{5}{u}^{5}-\\frac{1}{3}{u}^{3}+C\\hfill & \\hfill \\\\ \\hfill & =\\frac{1}{5}{\\cos}^{5}x-\\frac{1}{3}{\\cos}^{3}x+C.\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165043250980\" data-type=\"example\">\n<div id=\"fs-id1165043100234\" data-type=\"exercise\">\n<div id=\"fs-id1165042449637\" data-type=\"problem\">\n<div data-type=\"title\">\n<p><span style=\"font-size: 1rem; text-align: initial;\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{3}x{\\sin}^{2}xdx[\/latex].<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558894\">Hint<\/span><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043085127\" data-type=\"commentary\" data-element-type=\"hint\">\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1165042966824\" style=\"text-align: left;\">Write [latex]{\\cos}^{3}x={\\cos}^{2}x\\cos{x}=\\left(1-{\\sin}^{2}x\\right)\\cos{x}[\/latex] and let [latex]u=\\sin{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043248784\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165042980470\" data-type=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558895\">Show Solution<\/span><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043103949\" data-type=\"solution\">\n<p id=\"fs-id1165042190535\" style=\"text-align: center;\">[latex]\\frac{1}{3}{\\sin}^{3}x-\\frac{1}{5}{\\sin}^{5}x+C[\/latex]<\/p>\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<p>.<\/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=337&amp;end=450&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\/3.2TrigonometricIntegrals337to450_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<p><span style=\"font-size: 1rem; text-align: initial;\">In the next example, we see the strategy that must be applied when there are only even powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex]. For integrals of this type, the identities<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043066788\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)=\\frac{1-\\cos\\left(2x\\right)}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042530463\">and<\/p>\n<div id=\"fs-id1165043331511\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{\\cos}^{2}x=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(2x\\right)=\\frac{1+\\cos\\left(2x\\right)}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042368394\">are invaluable. These identities are sometimes known as <span class=\"no-emphasis\" data-type=\"term\"><em data-effect=\"italics\">power-reducing identities<\/em><\/span> and they may be derived from the double-angle identity [latex]\\cos\\left(2x\\right)={\\cos}^{2}x-{\\sin}^{2}x[\/latex] and the Pythagorean identity [latex]{\\cos}^{2}x+{\\sin}^{2}x=1[\/latex].<\/p>\n<div id=\"fs-id1165042936506\" data-type=\"example\">\n<div id=\"fs-id1165042318645\" data-type=\"exercise\">\n<div id=\"fs-id1165042632772\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>example:\u00a0Integrating an Even Power of [latex]\\sin{x}[\/latex]<\/h3>\n<div id=\"fs-id1165042632772\" data-type=\"problem\">\n<p id=\"fs-id1165042647060\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{2}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558892\">Show Solution<\/span><\/p>\n<div id=\"q44558892\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042965476\" data-type=\"solution\">\n<p id=\"fs-id1165043057271\">To evaluate this integral, let\u2019s use the trigonometric identity [latex]{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165043194450\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}{\\sin}^{2}xdx\\hfill & ={\\displaystyle\\int}\\left(\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)\\right)dx\\hfill \\\\ \\hfill & =\\frac{1}{2}x-\\frac{1}{4}\\sin\\left(2x\\right)+C.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165042936506\" data-type=\"example\">\n<div id=\"fs-id1165042318645\" data-type=\"exercise\">\n<div id=\"fs-id1165042632772\" data-type=\"problem\">\n<div data-type=\"title\">\n<p><span style=\"font-size: 1rem; text-align: initial;\">Evaluate [latex]\\displaystyle\\int {\\cos}^{2}xdx[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043096050\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165042989371\" data-type=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558890\">Hint<\/span><\/p>\n<div id=\"q44558890\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043272814\" data-type=\"commentary\" data-element-type=\"hint\">\n<div id=\"fs-id1165042375699\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]{\\cos}^{2}x=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(2x\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558891\">Show Solution<\/span><\/p>\n<div id=\"q44558891\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042982086\" data-type=\"solution\">\n<p id=\"fs-id1165042137283\">[latex]\\frac{1}{2}x+\\frac{1}{4}\\sin\\left(2x\\right)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm25554\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25554&theme=oea&iframe_resize_id=ohm25554&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"font-size: 1rem; text-align: initial;\">The general process for integrating products of powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex] is summarized in the following set of guidelines.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042604945\" 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 Products and Powers of sin <em data-effect=\"italics\">x<\/em> and cos <em data-effect=\"italics\">x<\/em><\/h3>\n<hr \/>\n<p id=\"fs-id1165042275182\">To integrate [latex]{\\displaystyle\\int}{\\cos}^{j}x{\\sin}^{k}xdx[\/latex] use the following strategies:<\/p>\n<ol id=\"fs-id1165043034337\" type=\"1\">\n<li>If [latex]k[\/latex] is odd, rewrite [latex]{\\sin}^{k}x={\\sin}^{k - 1}x\\sin{x}[\/latex] and use the identity [latex]{\\sin}^{2}x=1-{\\cos}^{2}x[\/latex] to rewrite [latex]{\\sin}^{k - 1}x[\/latex] in terms of [latex]\\cos{x}[\/latex]. Integrate using the substitution [latex]u=\\cos{x}[\/latex]. This substitution makes [latex]du=\\text{-}\\sin{x}dx[\/latex].<\/li>\n<li>If [latex]j[\/latex] is odd, rewrite [latex]{\\cos}^{j}x={\\cos}^{j - 1}x\\cos{x}[\/latex] and use the identity [latex]{\\cos}^{2}x=1-{\\sin}^{2}x[\/latex] to rewrite [latex]{\\cos}^{j - 1}x[\/latex] in terms of [latex]\\sin{x}[\/latex]. Integrate using the substitution [latex]u=\\sin{x}[\/latex]. This substitution makes [latex]du=\\cos{x}dx[\/latex]. (<em data-effect=\"italics\">Note<\/em>: If both [latex]j[\/latex] and [latex]k[\/latex] are odd, either strategy 1 or strategy 2 may be used.)<\/li>\n<li>If both [latex]j[\/latex] and [latex]k[\/latex] are even, use [latex]{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)[\/latex] and [latex]{\\cos}^{2}x=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(2x\\right)[\/latex]. After applying these formulas, simplify and reapply strategies 1 through 3 as appropriate.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043272282\" data-type=\"example\">\n<div id=\"fs-id1165043086427\" data-type=\"exercise\">\n<div id=\"fs-id1165043393170\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Integrating [latex]\\displaystyle\\int {\\cos}^{j}x{\\sin}^{k}xdx[\/latex] where <em data-effect=\"italics\">k<\/em> is Odd<\/h3>\n<div id=\"fs-id1165043393170\" data-type=\"problem\">\n<p id=\"fs-id1165042907373\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{8}x{\\sin}^{5}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558889\">Show Solution<\/span><\/p>\n<div id=\"q44558889\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043350731\" data-type=\"solution\">\n<p id=\"fs-id1165042505411\">Since the power on [latex]\\sin{x}[\/latex] is odd, use strategy 1. Thus,<\/p>\n<div id=\"fs-id1165042349814\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\cos}^{8}x{\\sin}^{5}xdx& ={\\displaystyle\\int}{\\cos}^{8}x{\\sin}^{4}x\\sin{x}dx\\hfill & & \\text{Break off }\\sin{x}.\\hfill \\\\ & ={\\displaystyle\\int}{\\cos}^{8}x{\\left({\\sin}^{2}x\\right)}^{2}\\sin{x}dx\\hfill & & \\text{Rewrite }{\\sin}^{4}x={\\left({\\sin}^{2}x\\right)}^{2}.\\hfill \\\\ & ={\\displaystyle\\int}{\\cos}^{8}x{\\left(1-{\\cos}^{2}x\\right)}^{2}\\sin{x}dx\\hfill & & \\text{Substitute }{\\sin}^{2}x=1-{\\cos}^{2}x.\\hfill \\\\ & ={\\displaystyle\\int}{u}^{8}{\\left(1-{u}^{2}\\right)}^{2}\\left(\\text{-}du\\right)\\hfill & & \\text{Let }u=\\cos{x}\\text{ and }du=\\text{-}\\sin{x}dx.\\hfill \\\\ & ={\\displaystyle\\int}\\left(\\text{-}{u}^{8}+2{u}^{10}-{u}^{12}\\right)du\\hfill & & \\text{Expand}.\\hfill \\\\ & =-\\frac{1}{9}{u}^{9}+\\frac{2}{11}{u}^{11}-\\frac{1}{13}{u}^{13}+C\\hfill & & \\text{Evaluate the integral}.\\hfill \\\\ & =-\\frac{1}{9}{\\cos}^{9}x+\\frac{2}{11}{\\cos}^{11}x-\\frac{1}{13}{\\cos}^{13}x+C.\\hfill & & \\text{Substitute }u=\\cos{x}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707196\" data-type=\"example\">\n<div id=\"fs-id1165042707198\" data-type=\"exercise\">\n<div id=\"fs-id1165042329519\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Integrating [latex]{\\displaystyle\\int}{\\cos}^{j}x{\\sin}^{k}xdx[\/latex] where <em data-effect=\"italics\">k<\/em> and <em data-effect=\"italics\">j<\/em> are Even<\/h3>\n<div id=\"fs-id1165042329519\" data-type=\"problem\">\n<p id=\"fs-id1165043179875\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{4}xdx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558879\">Show Solution<\/span><\/p>\n<div id=\"q44558879\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042832305\" data-type=\"solution\">\n<p id=\"fs-id1165042832307\">Since the power on [latex]\\sin{x}[\/latex] is even [latex]\\left(k=4\\right)[\/latex] and the power on [latex]\\cos{x}[\/latex] is even [latex]\\left(j=0\\right)[\/latex], we must use strategy 3. Thus,<\/p>\n<div id=\"fs-id1165043089888\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cccc}\\hfill {\\displaystyle\\int}{\\sin}^{4}xdx& ={\\displaystyle\\int}{\\left({\\sin}^{2}x\\right)}^{2}dx\\hfill & & \\text{Rewrite }{\\sin}^{4}x={\\left({\\sin}^{2}x\\right)}^{2}.\\hfill \\\\ & ={\\displaystyle\\int}{\\left(\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)\\right)}^{2}dx\\hfill & & \\text{Substitute }{\\sin}^{2}x=\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right).\\hfill \\\\ & ={\\displaystyle\\int}\\left(\\frac{1}{4}-\\frac{1}{2}\\cos\\left(2x\\right)+\\frac{1}{4}{\\cos}^{2}\\left(2x\\right)\\right)dx\\hfill & & \\text{Expand }{\\left(\\frac{1}{2}-\\frac{1}{2}\\cos\\left(2x\\right)\\right)}^{2}.\\hfill \\\\ & ={\\displaystyle\\int}\\left(\\frac{1}{4}-\\frac{1}{2}\\cos\\left(2x\\right)+\\frac{1}{4}\\left(\\frac{1}{2}+\\frac{1}{2}\\cos\\left(4x\\right)\\right)\\right)dx.\\hfill & & \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043074297\">Since [latex]{\\cos}^{2}\\left(2x\\right)[\/latex] has an even power, substitute [latex]{\\cos}^{2}\\left(2x\\right)=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(4x\\right)\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1165042508466\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}={\\displaystyle\\int}\\left(\\frac{3}{8}-\\frac{1}{2}\\cos\\left(2x\\right)+\\frac{1}{8}\\cos\\left(4x\\right)\\right)dx\\hfill & \\text{Simplify}.\\hfill \\\\ =\\frac{3}{8}x-\\frac{1}{4}\\sin\\left(2x\\right)+\\frac{1}{32}\\sin\\left(4x\\right)+C\\hfill & \\text{Evaluate the integral}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165042707196\" data-type=\"example\">\n<div id=\"fs-id1165042707198\" data-type=\"exercise\">\n<div id=\"fs-id1165042329519\" data-type=\"problem\">\n<div data-type=\"title\">\n<p><span style=\"font-size: 1rem; text-align: initial;\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{3}xdx[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043210017\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165042369121\" data-type=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558859\">Hint<\/span><\/p>\n<div id=\"q44558859\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043174075\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165042617688\">Use strategy 2. Write [latex]{\\cos}^{3}x={\\cos}^{2}x\\cos{x}[\/latex] and substitute [latex]{\\cos}^{2}x=1-{\\sin}^{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=\"q44558869\">Show Solution<\/span><\/p>\n<div id=\"q44558869\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043113739\" data-type=\"solution\">\n<p id=\"fs-id1165041795566\">[latex]\\sin{x}-\\frac{1}{3}{\\sin}^{3}x+C[\/latex]<\/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-id1165039562343\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165039562346\" data-type=\"exercise\">\n<div id=\"fs-id1165042518537\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165042518537\" data-type=\"problem\">\n<p id=\"fs-id1165042518539\">Evaluate [latex]{\\displaystyle\\int}{\\cos}^{2}\\left(3x\\right)dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558839\">Hint<\/span><\/p>\n<div id=\"q44558839\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042677393\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165043254305\">Use strategy 3. Substitute [latex]{\\cos}^{2}\\left(3x\\right)=\\frac{1}{2}+\\frac{1}{2}\\cos\\left(6x\\right)[\/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=\"q44558849\">Show Solution<\/span><\/p>\n<div id=\"q44558849\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042834041\" data-type=\"solution\">\n<p id=\"fs-id1165040796562\">[latex]\\frac{1}{2}x+\\frac{1}{12}\\sin\\left(6x\\right)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043131551\">In some areas of physics, such as quantum mechanics, signal processing, and the computation of Fourier series, it is often necessary to integrate products that include [latex]\\sin\\left(ax\\right)[\/latex], [latex]\\sin\\left(bx\\right)[\/latex], [latex]\\cos\\left(ax\\right)[\/latex], and [latex]\\cos\\left(bx\\right)[\/latex]. These integrals are evaluated by applying trigonometric identities, as outlined in the following rule.<\/p>\n<div id=\"fs-id1165042832676\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\"><strong>Rule: Integrating Products of Sines and Cosines of Different Angles<\/strong><\/h3>\n<hr \/>\n<p id=\"fs-id1165043248760\">To integrate products involving [latex]\\sin\\left(ax\\right)[\/latex], [latex]\\sin\\left(bx\\right)[\/latex], [latex]\\cos\\left(ax\\right)[\/latex], and [latex]\\cos\\left(bx\\right)[\/latex], use the substitutions<\/p>\n<div id=\"fs-id1165043229277\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\sin\\left(ax\\right)\\sin\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)-\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/div>\n<div id=\"fs-id1165042349512\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\sin\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\sin\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\sin\\left(\\left(a+b\\right)x\\right)[\/latex]<\/div>\n<div id=\"fs-id1165042332192\" class=\"numbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\cos\\left(ax\\right)\\cos\\left(bx\\right)=\\frac{1}{2}\\cos\\left(\\left(a-b\\right)x\\right)+\\frac{1}{2}\\cos\\left(\\left(a+b\\right)x\\right)[\/latex]<\/div>\n<p style=\"text-align: center;\">\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165043094088\">These formulas may be derived from the sum-of-angle formulas for sine and cosine.<\/p>\n<div id=\"fs-id1165043094091\" data-type=\"example\">\n<div id=\"fs-id1165043321357\" data-type=\"exercise\">\n<div id=\"fs-id1165043321359\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Evaluating [latex]\\displaystyle\\int \\sin\\left(ax\\right)\\cos\\left(bx\\right)dx[\/latex]<\/h3>\n<div id=\"fs-id1165043321359\" data-type=\"problem\">\n<p id=\"fs-id1165043272050\">Evaluate [latex]{\\displaystyle\\int}\\sin\\left(5x\\right)\\cos\\left(3x\\right)dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558799\">Show Solution<\/span><\/p>\n<div id=\"q44558799\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043181525\" data-type=\"solution\">\n<p id=\"fs-id1165043181527\">Apply the identity [latex]\\sin\\left(5x\\right)\\cos\\left(3x\\right)=\\frac{1}{2}\\sin\\left(2x\\right)+\\frac{1}{2}\\sin\\left(8x\\right)[\/latex]. Thus,<\/p>\n<div id=\"fs-id1165042349275\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}{\\displaystyle\\int}\\sin\\left(5x\\right)\\cos\\left(3x\\right)dx\\hfill & ={\\displaystyle\\int}\\frac{1}{2}\\sin\\left(2x\\right)+\\frac{1}{2}\\sin\\left(8x\\right)dx\\hfill \\\\ \\hfill & =-\\frac{1}{4}\\cos\\left(2x\\right)-\\frac{1}{16}\\cos\\left(8x\\right)+C.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042707106\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165043094213\" data-type=\"exercise\">\n<div id=\"fs-id1165043094215\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1165043094215\" data-type=\"problem\">\n<p id=\"fs-id1165043094217\">Evaluate [latex]{\\displaystyle\\int}\\cos\\left(6x\\right)\\cos\\left(5x\\right)dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558599\">Hint<\/span><\/p>\n<div id=\"q44558599\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042644355\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165042644361\">Substitute [latex]\\cos\\left(6x\\right)\\cos\\left(5x\\right)=\\frac{1}{2}\\cos{x}+\\frac{1}{2}\\cos\\left(11x\\right)[\/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=\"q44558699\">Show Solution<\/span><\/p>\n<div id=\"q44558699\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043395284\" data-type=\"solution\">\n<p id=\"fs-id1165040670224\">[latex]\\frac{1}{2}\\sin{x}+\\frac{1}{22}\\sin\\left(11x\\right)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\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-805\">\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":6,"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-805","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/805","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":26,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/805\/revisions"}],"predecessor-version":[{"id":2771,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/805\/revisions\/2771"}],"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\/805\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=805"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=805"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=805"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=805"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}