{"id":730,"date":"2025-06-20T17:08:03","date_gmt":"2025-06-20T17:08:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=730"},"modified":"2025-09-05T17:38:18","modified_gmt":"2025-09-05T17:38:18","slug":"other-strategies-for-integration-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/other-strategies-for-integration-fresh-take\/","title":{"raw":"Other Strategies for Integration: Fresh Take","rendered":"Other Strategies for Integration: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find integrals efficiently using an integral table<\/li>\r\n \t<li>Use technology to solve integration problems<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Tables of Integrals<\/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\">Integration tables are pre-computed lists of common integrals and their antiderivatives\u2014think of them as a mathematical reference library. They're incredibly useful for quickly evaluating integrals that match standard forms, but you need to know how to use them strategically.<\/p>\r\n<p class=\"whitespace-normal break-words\">To use tables effectively, you often need to massage your integral into the right form. This might involve:<\/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\"><strong>Substitution<\/strong> to match variables (like [latex]u = e^x[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Algebraic manipulation<\/strong> to match coefficients<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Completing the square<\/strong> for quadratic expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Factoring<\/strong> to reveal hidden patterns<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Don't panic if your answer looks different from the table\u2014correct antiderivatives can have multiple forms. For example:<\/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 \\frac{dx}{\\sqrt{1+x^2}} = \\ln(x+\\sqrt{x^2+1}) + C[\/latex] (trig substitution)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{dx}{\\sqrt{1+x^2}} = \\sinh^{-1}x + C[\/latex] (hyperbolic substitution)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">These are mathematically equivalent! Two antiderivatives differ only by a constant if they're both correct.<\/p>\r\n<p class=\"whitespace-normal break-words\">Tables work best when you understand the underlying techniques. Use them as a powerful tool in your integration toolkit, not as a replacement for learning fundamental methods. They're most effective when you can recognize which table formula applies and how to transform your integral to match it. When tables shine:<\/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\"><strong>Quick evaluation<\/strong> of integrals matching standard patterns<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Verification<\/strong> of your manual solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Pattern recognition<\/strong> for tackling similar problems<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Time-saving<\/strong> on exams or homework when you recognize a formula<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Computer Algebra Systems<\/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\">Computer Algebra Systems (CAS) like Wolfram Alpha, Mathematica, or advanced graphing calculators can solve complex integrals instantly through symbolic computation. They're incredibly powerful tools that can handle problems that would take humans significant time and effort\u2014but they work best when you understand what's happening behind the scenes.<\/p>\r\n<p class=\"whitespace-normal break-words\">Think of CAS as a powerful calculator, not a replacement for understanding. Use it to:<\/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\"><strong>Check your work<\/strong> after solving manually<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Handle computational heavy lifting<\/strong> once you understand the approach<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Explore complex problems<\/strong> that would be impractical by hand<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Understanding manual techniques helps you recognize when a CAS result makes sense and builds mathematical intuition. The tools solve problems instantly, but they can't always explain the reasoning\u2014that's where your knowledge of integration methods becomes invaluable.<\/p>\r\n<p class=\"whitespace-normal break-words\">When CAS Excels:<\/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\"><strong>Complex integrals<\/strong> that would be tedious to solve by hand<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Verification<\/strong> of your manual solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Time-sensitive situations<\/strong> like exams or homework deadlines<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Exploring patterns<\/strong> by trying multiple similar problems quickly<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox interact\" aria-label=\"Interact\">You can access <a href=\"https:\/\/www.integral-calculator.com\/\" target=\"_blank\" rel=\"noopener\">this integral calculator for more practice calculating integrals<\/a>.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165041932784\" data-type=\"problem\">\r\n<p id=\"fs-id1165042135957\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{3}xdx[\/latex] using a CAS. Compare the result to [latex]\\frac{1}{3}{\\cos}^{3}x-\\cos{x}+C[\/latex], the result we might have obtained using the technique for integrating odd powers of [latex]\\sin{x}[\/latex] discussed earlier in this chapter.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1165040740646\" data-type=\"solution\">\r\n<p id=\"fs-id1165041952449\">Using Wolfram Alpha, we obtain<\/p>\r\n\r\n<div id=\"fs-id1165041805693\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int {\\sin}^{3}xdx=\\frac{1}{12}\\left(\\cos\\left(3x\\right)-9\\cos{x}\\right)+C[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165040692062\">This looks quite different from [latex]\\frac{1}{3}{\\cos}^{3}x-\\cos{x}+C[\/latex]. To see that these antiderivatives are equivalent, we can make use of a few trigonometric identities:<\/p>\r\n\r\n<div id=\"fs-id1165041831576\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}\\hfill \\frac{1}{12}\\left(\\cos\\left(3x\\right)-9\\cos{x}\\right)&amp; =\\frac{1}{12}\\left(\\cos\\left(x+2x\\right)-9\\cos{x}\\right)\\hfill \\\\ &amp; =\\frac{1}{12}\\left(\\cos\\left(x\\right)\\cos\\left(2x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)-9\\cos{x}\\right)\\hfill \\\\ &amp; =\\frac{1}{12}\\left(\\cos{x}\\left(2{\\cos}^{2}x - 1\\right)-\\sin{x}\\left(2\\sin{x}\\cos{x}\\right)-9\\cos{x}\\right)\\hfill \\\\ &amp; =\\frac{1}{12}\\left(2{\\cos}^{3}x-\\cos{x} - 2\\cos{x}\\left(1-{\\cos}^{2}x\\right)-9\\cos{x}\\right)\\hfill \\\\ &amp; =\\frac{1}{12}\\left(4{\\cos}^{3}x - 12\\cos{x}\\right)\\hfill \\\\ &amp; =\\frac{1}{3}{\\cos}^{3}x-\\cos{x}.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042279232\">Thus, the two antiderivatives are identical.<\/p>\r\n<p id=\"fs-id1165041802085\">We may also use a CAS to compare the graphs of the two functions, as shown in the following figure.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_07_05_001\"><figcaption><\/figcaption>[caption id=\"\" align=\"aligncenter\" width=\"494\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233827\/CNX_Calc_Figure_07_05_001.jpg\" alt=\"This is the graph of a periodic function. The waves have an amplitude of approximately 0.7 and a period of approximately 10. The graph represents the functions y = cos^3(x)\/3 \u2013 cos(x) and y = 1\/12(cos(3x)-9cos(x). The graph is the same for both functions.\" width=\"494\" height=\"347\" data-media-type=\"image\/jpeg\" \/> Figure 1. The graphs of [latex]y=\\frac{1}{3}{\\cos}^{3}x-\\cos{x}[\/latex] and [latex]y=\\frac{1}{12}\\left(\\cos\\left(3x\\right)-9\\cos{x}\\right)[\/latex] are identical.[\/caption]<\/figure>\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 above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PfpNtoK41oE?controls=0&amp;start=233&amp;end=618&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\/3.5.2_233to618_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.5.2\" here (opens in new window)<\/a>.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165040797562\" data-type=\"problem\">\r\n<p id=\"fs-id1165040797565\">Use a CAS to evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}+4}}[\/latex].<\/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-id1165041836979\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165040745136\">Answers may vary.<\/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-id1165040744373\" data-type=\"solution\">\r\n<p id=\"fs-id1165042232195\">Possible solutions include [latex]{\\text{sinh}}^{-1}\\left(\\frac{x}{2}\\right)+C[\/latex] and [latex]\\text{ln}|\\sqrt{{x}^{2}+4}+x|+C[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find integrals efficiently using an integral table<\/li>\n<li>Use technology to solve integration problems<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Tables of Integrals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Integration tables are pre-computed lists of common integrals and their antiderivatives\u2014think of them as a mathematical reference library. They&#8217;re incredibly useful for quickly evaluating integrals that match standard forms, but you need to know how to use them strategically.<\/p>\n<p class=\"whitespace-normal break-words\">To use tables effectively, you often need to massage your integral into the right form. This might involve:<\/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\"><strong>Substitution<\/strong> to match variables (like [latex]u = e^x[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Algebraic manipulation<\/strong> to match coefficients<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Completing the square<\/strong> for quadratic expressions<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Factoring<\/strong> to reveal hidden patterns<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Don&#8217;t panic if your answer looks different from the table\u2014correct antiderivatives can have multiple forms. For example:<\/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 \\frac{dx}{\\sqrt{1+x^2}} = \\ln(x+\\sqrt{x^2+1}) + C[\/latex] (trig substitution)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{dx}{\\sqrt{1+x^2}} = \\sinh^{-1}x + C[\/latex] (hyperbolic substitution)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">These are mathematically equivalent! Two antiderivatives differ only by a constant if they&#8217;re both correct.<\/p>\n<p class=\"whitespace-normal break-words\">Tables work best when you understand the underlying techniques. Use them as a powerful tool in your integration toolkit, not as a replacement for learning fundamental methods. They&#8217;re most effective when you can recognize which table formula applies and how to transform your integral to match it. When tables shine:<\/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\"><strong>Quick evaluation<\/strong> of integrals matching standard patterns<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Verification<\/strong> of your manual solutions<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Pattern recognition<\/strong> for tackling similar problems<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Time-saving<\/strong> on exams or homework when you recognize a formula<\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Computer Algebra Systems<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Computer Algebra Systems (CAS) like Wolfram Alpha, Mathematica, or advanced graphing calculators can solve complex integrals instantly through symbolic computation. They&#8217;re incredibly powerful tools that can handle problems that would take humans significant time and effort\u2014but they work best when you understand what&#8217;s happening behind the scenes.<\/p>\n<p class=\"whitespace-normal break-words\">Think of CAS as a powerful calculator, not a replacement for understanding. Use it to:<\/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\"><strong>Check your work<\/strong> after solving manually<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Handle computational heavy lifting<\/strong> once you understand the approach<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Explore complex problems<\/strong> that would be impractical by hand<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Understanding manual techniques helps you recognize when a CAS result makes sense and builds mathematical intuition. The tools solve problems instantly, but they can&#8217;t always explain the reasoning\u2014that&#8217;s where your knowledge of integration methods becomes invaluable.<\/p>\n<p class=\"whitespace-normal break-words\">When CAS Excels:<\/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\"><strong>Complex integrals<\/strong> that would be tedious to solve by hand<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Verification<\/strong> of your manual solutions<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Time-sensitive situations<\/strong> like exams or homework deadlines<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Exploring patterns<\/strong> by trying multiple similar problems quickly<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox interact\" aria-label=\"Interact\">You can access <a href=\"https:\/\/www.integral-calculator.com\/\" target=\"_blank\" rel=\"noopener\">this integral calculator for more practice calculating integrals<\/a>.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165041932784\" data-type=\"problem\">\n<p id=\"fs-id1165042135957\">Evaluate [latex]{\\displaystyle\\int}{\\sin}^{3}xdx[\/latex] using a CAS. Compare the result to [latex]\\frac{1}{3}{\\cos}^{3}x-\\cos{x}+C[\/latex], the result we might have obtained using the technique for integrating odd powers of [latex]\\sin{x}[\/latex] discussed earlier in this chapter.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Show Solution<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165040740646\" data-type=\"solution\">\n<p id=\"fs-id1165041952449\">Using Wolfram Alpha, we obtain<\/p>\n<div id=\"fs-id1165041805693\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int {\\sin}^{3}xdx=\\frac{1}{12}\\left(\\cos\\left(3x\\right)-9\\cos{x}\\right)+C[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165040692062\">This looks quite different from [latex]\\frac{1}{3}{\\cos}^{3}x-\\cos{x}+C[\/latex]. To see that these antiderivatives are equivalent, we can make use of a few trigonometric identities:<\/p>\n<div id=\"fs-id1165041831576\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}\\hfill \\frac{1}{12}\\left(\\cos\\left(3x\\right)-9\\cos{x}\\right)& =\\frac{1}{12}\\left(\\cos\\left(x+2x\\right)-9\\cos{x}\\right)\\hfill \\\\ & =\\frac{1}{12}\\left(\\cos\\left(x\\right)\\cos\\left(2x\\right)-\\sin\\left(x\\right)\\sin\\left(2x\\right)-9\\cos{x}\\right)\\hfill \\\\ & =\\frac{1}{12}\\left(\\cos{x}\\left(2{\\cos}^{2}x - 1\\right)-\\sin{x}\\left(2\\sin{x}\\cos{x}\\right)-9\\cos{x}\\right)\\hfill \\\\ & =\\frac{1}{12}\\left(2{\\cos}^{3}x-\\cos{x} - 2\\cos{x}\\left(1-{\\cos}^{2}x\\right)-9\\cos{x}\\right)\\hfill \\\\ & =\\frac{1}{12}\\left(4{\\cos}^{3}x - 12\\cos{x}\\right)\\hfill \\\\ & =\\frac{1}{3}{\\cos}^{3}x-\\cos{x}.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042279232\">Thus, the two antiderivatives are identical.<\/p>\n<p id=\"fs-id1165041802085\">We may also use a CAS to compare the graphs of the two functions, as shown in the following figure.<\/p>\n<figure id=\"CNX_Calc_Figure_07_05_001\"><figcaption><\/figcaption><figure style=\"width: 494px\" 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\/11233827\/CNX_Calc_Figure_07_05_001.jpg\" alt=\"This is the graph of a periodic function. The waves have an amplitude of approximately 0.7 and a period of approximately 10. The graph represents the functions y = cos^3(x)\/3 \u2013 cos(x) and y = 1\/12(cos(3x)-9cos(x). The graph is the same for both functions.\" width=\"494\" height=\"347\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1. The graphs of [latex]y=\\frac{1}{3}{\\cos}^{3}x-\\cos{x}[\/latex] and [latex]y=\\frac{1}{12}\\left(\\cos\\left(3x\\right)-9\\cos{x}\\right)[\/latex] are identical.<\/figcaption><\/figure>\n<\/figure>\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 above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PfpNtoK41oE?controls=0&amp;start=233&amp;end=618&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\/3.5.2_233to618_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.5.2&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165040797562\" data-type=\"problem\">\n<p id=\"fs-id1165040797565\">Use a CAS to evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}+4}}[\/latex].<\/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-id1165041836979\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165040745136\">Answers may vary.<\/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-id1165040744373\" data-type=\"solution\">\n<p id=\"fs-id1165042232195\">Possible solutions include [latex]{\\text{sinh}}^{-1}\\left(\\frac{x}{2}\\right)+C[\/latex] and [latex]\\text{ln}|\\sqrt{{x}^{2}+4}+x|+C[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":29,"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\/730"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/730\/revisions"}],"predecessor-version":[{"id":2227,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/730\/revisions\/2227"}],"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\/730\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=730"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=730"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=730"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=730"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}