{"id":706,"date":"2025-06-20T17:06:40","date_gmt":"2025-06-20T17:06:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=706"},"modified":"2025-09-05T17:23:10","modified_gmt":"2025-09-05T17:23:10","slug":"trigonometric-integrals-fresh-take-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/trigonometric-integrals-fresh-take-2\/","title":{"raw":"Trigonometric Integrals: Fresh Take","rendered":"Trigonometric Integrals: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Integrate expressions containing products and powers of sine and cosine<\/li>\r\n \t<li>Integrate expressions containing products and powers of tangent and secant<\/li>\r\n \t<li>Use reduction formulas to simplify and solve trigonometric integrals<\/li>\r\n \t<li>Integrate expressions containing square roots of sums or differences of squares<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Integrating Products and Powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex]<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Integrating products and powers of sine and cosine might look intimidating, but there's a systematic game plan. The key insight is transforming these expressions into forms where u-substitution works perfectly\u2014either [latex]\\int \\sin^j x \\cos x\u00a0 dx[\/latex] or [latex]\\int \\cos^j x \\sin x\u00a0 dx[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategy 1 - Odd Power of Sine:<\/strong> If the power of [latex]\\sin x[\/latex] is odd, peel off one [latex]\\sin x[\/latex] and convert the rest using [latex]\\sin^2 x = 1 - \\cos^2 x[\/latex]. Then use [latex]u = \\cos x[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategy 2 - Odd Power of Cosine:<\/strong> If the power of [latex]\\cos x[\/latex] is odd, peel off one [latex]\\cos x[\/latex] and convert the rest using [latex]\\cos^2 x = 1 - \\sin^2 x[\/latex]. Then use [latex]u = \\sin x[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategy 3 - Both Powers Even:<\/strong> Use the power-reducing identities:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sin^2 x = \\frac{1 - \\cos(2x)}{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\cos^2 x = \\frac{1 + \\cos(2x)}{2}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Decision Tree:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Any odd powers?<\/strong> \u2192 Use Strategy 1 or 2<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Both even powers?<\/strong> \u2192 Use Strategy 3 (power-reducing identities)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Different angles?<\/strong> \u2192 Use product-to-sum identities<\/li>\r\n<\/ol>\r\nThese formulas cut even powers in half, making the integral manageable. You might need to apply them multiple times for higher powers.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<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\" data-mce-fragment=\"1\"><\/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<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section>\r\n<h2>Integrating Products and Powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex]<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Integrating products and powers of tangent and secant follows a similar playbook to sine and cosine, but with different strategic moves. Your goal is always the same: reshape the integral into either [latex]\\int \\tan^j x \\sec^2 x dx[\/latex] or [latex]\\int \\sec^j x \\sec x \\tan x dx[\/latex] so u-substitution can work its magic.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Your Strategic Game Plan:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategy 1 - Even Power of Secant:<\/strong> When the secant power is even (and [latex]\u2265 2[\/latex]), peel off [latex]\\sec^2 x[\/latex] and convert the remaining secants using [latex]\\sec^2 x = \\tan^2 x + 1[\/latex]. Use [latex]u = \\tan x[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategy 2 - Odd Power of Tangent:<\/strong> When the tangent power is odd (and secant power \u2265 1), peel off [latex]\\sec x \\tan x[\/latex] and convert remaining tangents using [latex]\\tan^2 x = \\sec^2 x - 1[\/latex]. Use [latex]u = \\sec x[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategy 3 - Odd Tangent, No Secant:<\/strong> For [latex]\\int \\tan^k x , dx[\/latex] where k is odd, rewrite as [latex]\\int \\tan^{k-2} x (\\sec^2 x - 1) , dx[\/latex] and split the integral.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategy 4 - Even Tangent, Odd Secant:<\/strong> Use [latex]\\tan^2 x = \\sec^2 x - 1[\/latex] to convert everything to secants, then apply integration by parts for odd secant powers.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Essential Identity Arsenal:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan^2 x + 1 = \\sec^2 x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan^2 x = \\sec^2 x - 1[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The \"peel off\" strategy works because [latex]\\sec^2 x[\/latex] is the derivative of [latex]\\tan x[\/latex], and [latex]\\sec x \\tan x[\/latex] is the derivative of [latex]\\sec x[\/latex]. This makes [latex]u[\/latex]-substitution clean and efficient.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox interact\" aria-label=\"Interact\">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>.<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<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\" data-mce-fragment=\"1\"><\/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>\r\n<h2 data-type=\"title\">Reduction Formulas<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Think of reduction formulas as a \"power ladder\" that helps you climb down from high powers to manageable ones. When you encounter [latex]\\int \\sec^5 x , dx[\/latex] or [latex]\\int \\tan^6 x , dx[\/latex], these formulas systematically reduce the power until you reach integrals you can actually solve. Each formula trades your current integral for a simpler one with a power that's 2 steps lower. You keep applying the formula until you reach something basic like [latex]\\int \\sec x , dx[\/latex] or [latex]\\int 1 , dx[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Power-Reducing Formulas:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For secant: [latex]\\int \\sec^n x , dx = \\frac{1}{n-1}\\sec^{n-2}x \\tan x + \\frac{n-2}{n-1}\\int \\sec^{n-2}x , dx[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">For tangent: [latex]\\int \\tan^n x , dx = \\frac{1}{n-1}\\tan^{n-1}x - \\int \\tan^{n-2}x , dx[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Apply the formula<\/strong> to reduce the power<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Repeat as needed<\/strong> until you hit a base case you know how to integrate<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Work backwards<\/strong> through your chain of reductions<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Base Cases to Remember:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\int \\sec x , dx = \\ln|\\sec x + \\tan x| + C[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\int \\tan^0 x , dx = \\int 1 , dx = x + C[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\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<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<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\" data-mce-fragment=\"1\"><\/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<h2 data-type=\"title\">Integrals Involving [latex]\\sqrt{{a}^{2}-{x}^{2}}[\/latex]<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">When you encounter [latex]\\int \\sqrt{a^2 - x^2} , dx[\/latex], traditional methods hit a wall. But here's the clever insight: this expression has the same form as the Pythagorean identity [latex]\\sin^2\\theta + \\cos^2\\theta = 1[\/latex]. By using trigonometric substitution, you can transform an impossible-looking integral into familiar trigonometric integration.<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\sqrt{a^2 - x^2}[\/latex], use [latex]x = a\\sin\\theta[\/latex] and [latex]dx = a\\cos\\theta , d\\theta[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This transforms: [latex]\\sqrt{a^2 - x^2} = \\sqrt{a^2 - a^2\\sin^2\\theta} = \\sqrt{a^2(1-\\sin^2\\theta)} = \\sqrt{a^2\\cos^2\\theta} = a\\cos\\theta[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Check first<\/strong>: Can you solve this with u-substitution or a basic formula? (Save trig substitution for when you really need it)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Make the substitution<\/strong>: [latex]x = a\\sin\\theta[\/latex], [latex]dx = a\\cos\\theta , d\\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Simplify the square root<\/strong>: [latex]\\sqrt{a^2 - x^2} = a\\cos\\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Integrate<\/strong>: Use your trigonometric integration techniques<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Convert back<\/strong>: Build a reference triangle to express your answer in terms of [latex]x[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>The Reference Triangle Method:<\/strong> Since [latex]x = a\\sin\\theta[\/latex], you know [latex]\\sin\\theta = \\frac{x}{a}[\/latex]. Draw a right triangle with:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Hypotenuse: [latex]a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Opposite side: [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Adjacent side: [latex]\\sqrt{a^2 - x^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Therefore: [latex]\\theta = \\sin^{-1}\\left(\\frac{x}{a}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165042073730\" data-type=\"problem\">\r\n<p id=\"fs-id1165042004532\">Evaluate [latex]\\displaystyle\\int \\frac{\\sqrt{4-{x}^{2}}}{x}dx[\/latex].<\/p>\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-id1165040797018\" data-type=\"solution\">\r\n<p id=\"fs-id1165041759427\">First make the substitutions [latex]x=2\\sin\\theta [\/latex] and [latex]dx=2\\cos\\theta d\\theta [\/latex]. Since [latex]\\sin\\theta =\\frac{x}{2}[\/latex], we can construct the reference triangle shown in the following figure.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_07_03_003\"><figcaption><\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"380\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233803\/CNX_Calc_Figure_07_03_003.jpg\" alt=\"This figure is a right triangle. It has an angle labeled theta. This angle is opposite the vertical side. The vertical leg is labeled x, and the horizontal leg is labeled as the square root of (4 \u2013 x^2). To the left of the triangle is the equation sin(theta) = x\/2.\" width=\"380\" height=\"177\" data-media-type=\"image\/jpeg\" \/> Figure 3. A reference triangle can be constructed for this example.[\/caption]<\/figure>\r\n<p id=\"fs-id1165040739936\">Thus,<\/p>\r\n\r\n<div id=\"fs-id1165040697663\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccccc}\\hfill {\\displaystyle\\int \\frac{\\sqrt{4-{x}^{2}}}{x}dx}&amp; ={\\displaystyle\\int \\frac{\\sqrt{4-{\\left(2\\sin\\theta \\right)}^{2}}}{2\\sin\\theta }2\\cos\\theta d\\theta} \\hfill &amp; &amp; &amp; \\text{Substitute}x=2\\sin\\theta \\text{and}=2\\cos\\theta d\\theta .\\hfill \\\\ &amp; ={\\displaystyle\\int \\frac{2{\\cos}^{2}\\theta }{\\sin\\theta }d\\theta} \\hfill &amp; &amp; &amp; \\text{Substitute}{\\cos}^{2}\\theta =1-{\\sin}^{2}\\theta \\text{and simplify.}\\hfill \\\\ &amp; ={\\displaystyle\\int \\frac{2\\left(1-{\\sin}^{2}\\theta \\right)}{\\sin\\theta }d\\theta} \\hfill &amp; &amp; &amp; \\text{Substitute}{\\sin}^{2}\\theta =1-{\\cos}^{2}\\theta .\\hfill \\\\ &amp; ={\\displaystyle\\int \\left(2\\csc\\theta -2\\sin\\theta \\right)d\\theta} \\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Separate the numerator, simplify, and use}\\hfill \\\\ \\csc\\theta =\\frac{1}{\\sin\\theta }.\\hfill \\end{array}\\hfill \\\\ &amp; =2\\text{ln}|\\csc\\theta -\\cot\\theta |+2\\cos\\theta +C\\hfill &amp; &amp; &amp; \\text{Evaluate the integral.}\\hfill \\\\ &amp; =2\\text{ln}|\\frac{2}{x}-\\frac{\\sqrt{4-{x}^{2}}}{x}|+\\sqrt{4-{x}^{2}}+C.\\hfill &amp; &amp; &amp; \\begin{array}{c}\\text{Use the reference triangle to rewrite the}\\hfill \\\\ \\text{expression in terms of}x\\text{and simplify.}\\hfill \\end{array}\\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165042033160\" data-type=\"problem\">\r\n<p id=\"fs-id1165042033163\">Rewrite the integral [latex]\\displaystyle\\int \\frac{{x}^{3}}{\\sqrt{25-{x}^{2}}}dx[\/latex] using the appropriate trigonometric substitution (do not evaluate the integral).<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558895\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1165041841595\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165041951969\">Substitute [latex]x=5\\sin\\theta [\/latex] and [latex]dx=5\\cos\\theta d\\theta [\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558896\"]\r\n<div id=\"fs-id1165042237758\" data-type=\"solution\">\r\n<p id=\"fs-id1165040640415\">[latex]{\\displaystyle\\int }^{\\text{ }}125{\\sin}^{3}\\theta d\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/bCIrhw0sjlU?controls=0&amp;start=1071&amp;end=1182&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/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\/3.3TrigonometricSubstitution1071to1182_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Trigonometric Substitution\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Integrating Expressions Involving [latex]\\sqrt{{a}^{2}+{x}^{2}}[\/latex]<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Unlike [latex]\\sqrt{a^2 - x^2}[\/latex] which has a limited domain, [latex]\\sqrt{a^2 + x^2}[\/latex] is defined for all real values of [latex]x[\/latex]. This changes everything about our substitution strategy. We need a trigonometric function that can also take on any real value\u2014and that's where tangent comes in.<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\sqrt{a^2 + x^2}[\/latex], use [latex]x = a\\tan\\theta[\/latex] and [latex]dx = a\\sec^2\\theta , d\\theta[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This transforms: [latex]\\sqrt{a^2 + x^2} = \\sqrt{a^2 + a^2\\tan^2\\theta} = \\sqrt{a^2(1 + \\tan^2\\theta)} = \\sqrt{a^2\\sec^2\\theta} = a\\sec\\theta[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Why This Works:<\/strong> The magic lies in the identity [latex]1 + \\tan^2\\theta = \\sec^2\\theta[\/latex]. Since we restrict [latex]\\theta[\/latex] to [latex](-\\frac{\\pi}{2}, \\frac{\\pi}{2})[\/latex], we have [latex]\\sec\\theta &gt; 0[\/latex], so [latex]\\sqrt{\\sec^2\\theta} = \\sec\\theta[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Your Reference Triangle Setup:<\/strong> Since [latex]x = a\\tan\\theta[\/latex], you know [latex]\\tan\\theta = \\frac{x}{a}[\/latex]. Draw a right triangle with:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Adjacent side: [latex]a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Opposite side: [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Hypotenuse: [latex]\\sqrt{a^2 + x^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Therefore: [latex]\\theta = \\tan^{-1}\\left(\\frac{x}{a}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165042041810\" data-type=\"problem\">\r\n<p id=\"fs-id1165042041812\">Rewrite [latex]{\\displaystyle\\int }^{\\text{ }}{x}^{3}\\sqrt{{x}^{2}+4}dx[\/latex] by using a substitution involving [latex]\\tan\\theta [\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558890\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558890\"]\r\n<div id=\"fs-id1165041921699\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165041921706\">Use [latex]x=2\\tan\\theta [\/latex] and [latex]dx=2{\\sec}^{2}\\theta d\\theta [\/latex].<\/p>\r\n\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-id1165041952815\" data-type=\"solution\">\r\n<p id=\"fs-id1165041952817\">[latex]{\\displaystyle\\int }^{\\text{ }}32{\\tan}^{3}\\theta {\\sec}^{3}\\theta d\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/bCIrhw0sjlU?controls=0&amp;start=1688&amp;end=1797&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/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\/3.3TrigonometricSubstitution1688to1797_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Trigonometric Substitution\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Integrating Expressions Involving [latex]\\sqrt{{x}^{2}-{a}^{2}}[\/latex]<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">The expression [latex]\\sqrt{x^2 - a^2}[\/latex] is the most challenging of the three trigonometric substitution cases because of its split domain. Unlike the previous cases, this square root only exists when [latex]x \\leq -a[\/latex] or [latex]x \\geq a[\/latex]. This creates a sign issue that requires extra care.<\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\sqrt{x^2 - a^2}[\/latex], use [latex]x = a\\sec\\theta[\/latex] and [latex]dx = a\\sec\\theta \\tan\\theta , d\\theta[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This transforms: [latex]\\sqrt{x^2 - a^2} = \\sqrt{a^2\\sec^2\\theta - a^2} = \\sqrt{a^2(\\sec^2\\theta - 1)} = \\sqrt{a^2\\tan^2\\theta} = |a\\tan\\theta|[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Critical Sign Issue:<\/strong> The absolute value [latex]|a\\tan\\theta|[\/latex] depends on where you are in the domain:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]x \\geq a[\/latex]: [latex]|a\\tan\\theta| = a\\tan\\theta[\/latex] (positive)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For [latex]x \\leq -a[\/latex]: [latex]|a\\tan\\theta| = -a\\tan\\theta[\/latex] (negative)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Your Reference Triangle Strategy:<\/strong> Since [latex]x = a\\sec\\theta[\/latex], you know [latex]\\sec\\theta = \\frac{x}{a}[\/latex]. You need different setups for each part of the domain, but the basic triangle has:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Hypotenuse: [latex]|x|[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Adjacent side: [latex]a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Opposite side: [latex]\\sqrt{x^2 - a^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Therefore: [latex]\\theta = \\sec^{-1}\\left(\\frac{x}{a}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Always check which part of the domain you're working in, especially for definite integrals. The sign of [latex]\\tan\\theta[\/latex] will determine whether you need the positive or negative version of your final answer.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165041757013\" data-type=\"problem\">\r\n<p id=\"fs-id1165041757015\">Evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}[\/latex]. Assume that [latex]x&gt;2[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558869\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558869\"]\r\n<div id=\"fs-id1165040743205\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165040743213\">Substitute [latex]x=2\\sec\\theta [\/latex] and [latex]dx=2\\sec\\theta \\tan\\theta d\\theta [\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558879\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558879\"]\r\n<div id=\"fs-id1165042272930\" data-type=\"solution\">\r\n<p id=\"fs-id1165042272932\">[latex]\\text{ln}|\\frac{x}{2}+\\frac{\\sqrt{{x}^{2}-4}}{2}|+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/bCIrhw0sjlU?controls=0&amp;start=2177&amp;end=2312&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/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\/3.3TrigonometricSubstitution2177to2312_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.3 Trigonometric Substitution\" here (opens in new window)<\/a>.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Integrate expressions containing products and powers of sine and cosine<\/li>\n<li>Integrate expressions containing products and powers of tangent and secant<\/li>\n<li>Use reduction formulas to simplify and solve trigonometric integrals<\/li>\n<li>Integrate expressions containing square roots of sums or differences of squares<\/li>\n<\/ul>\n<\/section>\n<h2>Integrating Products and Powers of [latex]\\sin{x}[\/latex] and [latex]\\cos{x}[\/latex]<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Integrating products and powers of sine and cosine might look intimidating, but there&#8217;s a systematic game plan. The key insight is transforming these expressions into forms where u-substitution works perfectly\u2014either [latex]\\int \\sin^j x \\cos x\u00a0 dx[\/latex] or [latex]\\int \\cos^j x \\sin x\u00a0 dx[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategy 1 &#8211; Odd Power of Sine:<\/strong> If the power of [latex]\\sin x[\/latex] is odd, peel off one [latex]\\sin x[\/latex] and convert the rest using [latex]\\sin^2 x = 1 - \\cos^2 x[\/latex]. Then use [latex]u = \\cos x[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategy 2 &#8211; Odd Power of Cosine:<\/strong> If the power of [latex]\\cos x[\/latex] is odd, peel off one [latex]\\cos x[\/latex] and convert the rest using [latex]\\cos^2 x = 1 - \\sin^2 x[\/latex]. Then use [latex]u = \\sin x[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategy 3 &#8211; Both Powers Even:<\/strong> Use the power-reducing identities:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\sin^2 x = \\frac{1 - \\cos(2x)}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\cos^2 x = \\frac{1 + \\cos(2x)}{2}[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Decision Tree:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Any odd powers?<\/strong> \u2192 Use Strategy 1 or 2<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Both even powers?<\/strong> \u2192 Use Strategy 3 (power-reducing identities)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Different angles?<\/strong> \u2192 Use product-to-sum identities<\/li>\n<\/ol>\n<p>These formulas cut even powers in half, making the integral manageable. You might need to apply them multiple times for higher powers.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558894\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558895\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/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\" data-mce-fragment=\"1\"><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558890\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558891\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558859\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558869\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558839\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558849\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558599\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558699\">Show Solution<\/button><\/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<\/section>\n<h2>Integrating Products and Powers of [latex]\\tan{x}[\/latex] and [latex]\\sec{x}[\/latex]<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Integrating products and powers of tangent and secant follows a similar playbook to sine and cosine, but with different strategic moves. Your goal is always the same: reshape the integral into either [latex]\\int \\tan^j x \\sec^2 x dx[\/latex] or [latex]\\int \\sec^j x \\sec x \\tan x dx[\/latex] so u-substitution can work its magic.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Your Strategic Game Plan:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategy 1 &#8211; Even Power of Secant:<\/strong> When the secant power is even (and [latex]\u2265 2[\/latex]), peel off [latex]\\sec^2 x[\/latex] and convert the remaining secants using [latex]\\sec^2 x = \\tan^2 x + 1[\/latex]. Use [latex]u = \\tan x[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategy 2 &#8211; Odd Power of Tangent:<\/strong> When the tangent power is odd (and secant power \u2265 1), peel off [latex]\\sec x \\tan x[\/latex] and convert remaining tangents using [latex]\\tan^2 x = \\sec^2 x - 1[\/latex]. Use [latex]u = \\sec x[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategy 3 &#8211; Odd Tangent, No Secant:<\/strong> For [latex]\\int \\tan^k x , dx[\/latex] where k is odd, rewrite as [latex]\\int \\tan^{k-2} x (\\sec^2 x - 1) , dx[\/latex] and split the integral.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategy 4 &#8211; Even Tangent, Odd Secant:<\/strong> Use [latex]\\tan^2 x = \\sec^2 x - 1[\/latex] to convert everything to secants, then apply integration by parts for odd secant powers.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Essential Identity Arsenal:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\tan^2 x + 1 = \\sec^2 x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan^2 x = \\sec^2 x - 1[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The &#8220;peel off&#8221; strategy works because [latex]\\sec^2 x[\/latex] is the derivative of [latex]\\tan x[\/latex], and [latex]\\sec x \\tan x[\/latex] is the derivative of [latex]\\sec x[\/latex]. This makes [latex]u[\/latex]-substitution clean and efficient.<\/p>\n<\/div>\n<section class=\"textbox interact\" aria-label=\"Interact\">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>.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44557899\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44554899\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44555899\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/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\" data-mce-fragment=\"1\"><\/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<h2 data-type=\"title\">Reduction Formulas<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of reduction formulas as a &#8220;power ladder&#8221; that helps you climb down from high powers to manageable ones. When you encounter [latex]\\int \\sec^5 x , dx[\/latex] or [latex]\\int \\tan^6 x , dx[\/latex], these formulas systematically reduce the power until you reach integrals you can actually solve. Each formula trades your current integral for a simpler one with a power that&#8217;s 2 steps lower. You keep applying the formula until you reach something basic like [latex]\\int \\sec x , dx[\/latex] or [latex]\\int 1 , dx[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Power-Reducing Formulas:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For secant: [latex]\\int \\sec^n x , dx = \\frac{1}{n-1}\\sec^{n-2}x \\tan x + \\frac{n-2}{n-1}\\int \\sec^{n-2}x , dx[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">For tangent: [latex]\\int \\tan^n x , dx = \\frac{1}{n-1}\\tan^{n-1}x - \\int \\tan^{n-2}x , dx[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Apply the formula<\/strong> to reduce the power<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Repeat as needed<\/strong> until you hit a base case you know how to integrate<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Work backwards<\/strong> through your chain of reductions<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Base Cases to Remember:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\int \\sec x , dx = \\ln|\\sec x + \\tan x| + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\tan^0 x , dx = \\int 1 , dx = x + C[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44550899\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44551899\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/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\" data-mce-fragment=\"1\"><\/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<h2 data-type=\"title\">Integrals Involving [latex]\\sqrt{{a}^{2}-{x}^{2}}[\/latex]<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When you encounter [latex]\\int \\sqrt{a^2 - x^2} , dx[\/latex], traditional methods hit a wall. But here&#8217;s the clever insight: this expression has the same form as the Pythagorean identity [latex]\\sin^2\\theta + \\cos^2\\theta = 1[\/latex]. By using trigonometric substitution, you can transform an impossible-looking integral into familiar trigonometric integration.<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\sqrt{a^2 - x^2}[\/latex], use [latex]x = a\\sin\\theta[\/latex] and [latex]dx = a\\cos\\theta , d\\theta[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This transforms: [latex]\\sqrt{a^2 - x^2} = \\sqrt{a^2 - a^2\\sin^2\\theta} = \\sqrt{a^2(1-\\sin^2\\theta)} = \\sqrt{a^2\\cos^2\\theta} = a\\cos\\theta[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Check first<\/strong>: Can you solve this with u-substitution or a basic formula? (Save trig substitution for when you really need it)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Make the substitution<\/strong>: [latex]x = a\\sin\\theta[\/latex], [latex]dx = a\\cos\\theta , d\\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Simplify the square root<\/strong>: [latex]\\sqrt{a^2 - x^2} = a\\cos\\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Integrate<\/strong>: Use your trigonometric integration techniques<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Convert back<\/strong>: Build a reference triangle to express your answer in terms of [latex]x[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>The Reference Triangle Method:<\/strong> Since [latex]x = a\\sin\\theta[\/latex], you know [latex]\\sin\\theta = \\frac{x}{a}[\/latex]. Draw a right triangle with:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Hypotenuse: [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Opposite side: [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Adjacent side: [latex]\\sqrt{a^2 - x^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Therefore: [latex]\\theta = \\sin^{-1}\\left(\\frac{x}{a}\\right)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165042073730\" data-type=\"problem\">\n<p id=\"fs-id1165042004532\">Evaluate [latex]\\displaystyle\\int \\frac{\\sqrt{4-{x}^{2}}}{x}dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Show Solution<\/button><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165040797018\" data-type=\"solution\">\n<p id=\"fs-id1165041759427\">First make the substitutions [latex]x=2\\sin\\theta[\/latex] and [latex]dx=2\\cos\\theta d\\theta[\/latex]. Since [latex]\\sin\\theta =\\frac{x}{2}[\/latex], we can construct the reference triangle shown in the following figure.<\/p>\n<figure id=\"CNX_Calc_Figure_07_03_003\"><figcaption><\/figcaption><figure style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233803\/CNX_Calc_Figure_07_03_003.jpg\" alt=\"This figure is a right triangle. It has an angle labeled theta. This angle is opposite the vertical side. The vertical leg is labeled x, and the horizontal leg is labeled as the square root of (4 \u2013 x^2). To the left of the triangle is the equation sin(theta) = x\/2.\" width=\"380\" height=\"177\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 3. A reference triangle can be constructed for this example.<\/figcaption><\/figure>\n<\/figure>\n<p id=\"fs-id1165040739936\">Thus,<\/p>\n<div id=\"fs-id1165040697663\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccccc}\\hfill {\\displaystyle\\int \\frac{\\sqrt{4-{x}^{2}}}{x}dx}& ={\\displaystyle\\int \\frac{\\sqrt{4-{\\left(2\\sin\\theta \\right)}^{2}}}{2\\sin\\theta }2\\cos\\theta d\\theta} \\hfill & & & \\text{Substitute}x=2\\sin\\theta \\text{and}=2\\cos\\theta d\\theta .\\hfill \\\\ & ={\\displaystyle\\int \\frac{2{\\cos}^{2}\\theta }{\\sin\\theta }d\\theta} \\hfill & & & \\text{Substitute}{\\cos}^{2}\\theta =1-{\\sin}^{2}\\theta \\text{and simplify.}\\hfill \\\\ & ={\\displaystyle\\int \\frac{2\\left(1-{\\sin}^{2}\\theta \\right)}{\\sin\\theta }d\\theta} \\hfill & & & \\text{Substitute}{\\sin}^{2}\\theta =1-{\\cos}^{2}\\theta .\\hfill \\\\ & ={\\displaystyle\\int \\left(2\\csc\\theta -2\\sin\\theta \\right)d\\theta} \\hfill & & & \\begin{array}{c}\\text{Separate the numerator, simplify, and use}\\hfill \\\\ \\csc\\theta =\\frac{1}{\\sin\\theta }.\\hfill \\end{array}\\hfill \\\\ & =2\\text{ln}|\\csc\\theta -\\cot\\theta |+2\\cos\\theta +C\\hfill & & & \\text{Evaluate the integral.}\\hfill \\\\ & =2\\text{ln}|\\frac{2}{x}-\\frac{\\sqrt{4-{x}^{2}}}{x}|+\\sqrt{4-{x}^{2}}+C.\\hfill & & & \\begin{array}{c}\\text{Use the reference triangle to rewrite the}\\hfill \\\\ \\text{expression in terms of}x\\text{and simplify.}\\hfill \\end{array}\\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165042033160\" data-type=\"problem\">\n<p id=\"fs-id1165042033163\">Rewrite the integral [latex]\\displaystyle\\int \\frac{{x}^{3}}{\\sqrt{25-{x}^{2}}}dx[\/latex] using the appropriate trigonometric substitution (do not evaluate the integral).<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558895\">Hint<\/button><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041841595\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165041951969\">Substitute [latex]x=5\\sin\\theta[\/latex] and [latex]dx=5\\cos\\theta d\\theta[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558896\">Show Solution<\/button><\/p>\n<div id=\"q44558896\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042237758\" data-type=\"solution\">\n<p id=\"fs-id1165040640415\">[latex]{\\displaystyle\\int }^{\\text{ }}125{\\sin}^{3}\\theta d\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/bCIrhw0sjlU?controls=0&amp;start=1071&amp;end=1182&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/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\/3.3TrigonometricSubstitution1071to1182_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Trigonometric Substitution&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h2 data-type=\"title\">Integrating Expressions Involving [latex]\\sqrt{{a}^{2}+{x}^{2}}[\/latex]<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Unlike [latex]\\sqrt{a^2 - x^2}[\/latex] which has a limited domain, [latex]\\sqrt{a^2 + x^2}[\/latex] is defined for all real values of [latex]x[\/latex]. This changes everything about our substitution strategy. We need a trigonometric function that can also take on any real value\u2014and that&#8217;s where tangent comes in.<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\sqrt{a^2 + x^2}[\/latex], use [latex]x = a\\tan\\theta[\/latex] and [latex]dx = a\\sec^2\\theta , d\\theta[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This transforms: [latex]\\sqrt{a^2 + x^2} = \\sqrt{a^2 + a^2\\tan^2\\theta} = \\sqrt{a^2(1 + \\tan^2\\theta)} = \\sqrt{a^2\\sec^2\\theta} = a\\sec\\theta[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Why This Works:<\/strong> The magic lies in the identity [latex]1 + \\tan^2\\theta = \\sec^2\\theta[\/latex]. Since we restrict [latex]\\theta[\/latex] to [latex](-\\frac{\\pi}{2}, \\frac{\\pi}{2})[\/latex], we have [latex]\\sec\\theta > 0[\/latex], so [latex]\\sqrt{\\sec^2\\theta} = \\sec\\theta[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Your Reference Triangle Setup:<\/strong> Since [latex]x = a\\tan\\theta[\/latex], you know [latex]\\tan\\theta = \\frac{x}{a}[\/latex]. Draw a right triangle with:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Adjacent side: [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Opposite side: [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Hypotenuse: [latex]\\sqrt{a^2 + x^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Therefore: [latex]\\theta = \\tan^{-1}\\left(\\frac{x}{a}\\right)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165042041810\" data-type=\"problem\">\n<p id=\"fs-id1165042041812\">Rewrite [latex]{\\displaystyle\\int }^{\\text{ }}{x}^{3}\\sqrt{{x}^{2}+4}dx[\/latex] by using a substitution involving [latex]\\tan\\theta[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558890\">Hint<\/button><\/p>\n<div id=\"q44558890\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041921699\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165041921706\">Use [latex]x=2\\tan\\theta[\/latex] and [latex]dx=2{\\sec}^{2}\\theta d\\theta[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558891\">Show Solution<\/button><\/p>\n<div id=\"q44558891\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041952815\" data-type=\"solution\">\n<p id=\"fs-id1165041952817\">[latex]{\\displaystyle\\int }^{\\text{ }}32{\\tan}^{3}\\theta {\\sec}^{3}\\theta d\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/bCIrhw0sjlU?controls=0&amp;start=1688&amp;end=1797&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/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\/3.3TrigonometricSubstitution1688to1797_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Trigonometric Substitution&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h2 data-type=\"title\">Integrating Expressions Involving [latex]\\sqrt{{x}^{2}-{a}^{2}}[\/latex]<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The expression [latex]\\sqrt{x^2 - a^2}[\/latex] is the most challenging of the three trigonometric substitution cases because of its split domain. Unlike the previous cases, this square root only exists when [latex]x \\leq -a[\/latex] or [latex]x \\geq a[\/latex]. This creates a sign issue that requires extra care.<\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\sqrt{x^2 - a^2}[\/latex], use [latex]x = a\\sec\\theta[\/latex] and [latex]dx = a\\sec\\theta \\tan\\theta , d\\theta[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This transforms: [latex]\\sqrt{x^2 - a^2} = \\sqrt{a^2\\sec^2\\theta - a^2} = \\sqrt{a^2(\\sec^2\\theta - 1)} = \\sqrt{a^2\\tan^2\\theta} = |a\\tan\\theta|[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Critical Sign Issue:<\/strong> The absolute value [latex]|a\\tan\\theta|[\/latex] depends on where you are in the domain:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">For [latex]x \\geq a[\/latex]: [latex]|a\\tan\\theta| = a\\tan\\theta[\/latex] (positive)<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]x \\leq -a[\/latex]: [latex]|a\\tan\\theta| = -a\\tan\\theta[\/latex] (negative)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Your Reference Triangle Strategy:<\/strong> Since [latex]x = a\\sec\\theta[\/latex], you know [latex]\\sec\\theta = \\frac{x}{a}[\/latex]. You need different setups for each part of the domain, but the basic triangle has:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Hypotenuse: [latex]|x|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Adjacent side: [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Opposite side: [latex]\\sqrt{x^2 - a^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Therefore: [latex]\\theta = \\sec^{-1}\\left(\\frac{x}{a}\\right)[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Always check which part of the domain you&#8217;re working in, especially for definite integrals. The sign of [latex]\\tan\\theta[\/latex] will determine whether you need the positive or negative version of your final answer.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165041757013\" data-type=\"problem\">\n<p id=\"fs-id1165041757015\">Evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}[\/latex]. Assume that [latex]x>2[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558869\">Hint<\/button><\/p>\n<div id=\"q44558869\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165040743205\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165040743213\">Substitute [latex]x=2\\sec\\theta[\/latex] and [latex]dx=2\\sec\\theta \\tan\\theta d\\theta[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558879\">Show Solution<\/button><\/p>\n<div id=\"q44558879\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042272930\" data-type=\"solution\">\n<p id=\"fs-id1165042272932\">[latex]\\text{ln}|\\frac{x}{2}+\\frac{\\sqrt{{x}^{2}-4}}{2}|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/bCIrhw0sjlU?controls=0&amp;start=2177&amp;end=2312&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/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\/3.3TrigonometricSubstitution2177to2312_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.3 Trigonometric Substitution&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":541,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/706"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/706\/revisions"}],"predecessor-version":[{"id":2225,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/706\/revisions\/2225"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/541"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/706\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=706"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=706"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=706"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=706"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}