{"id":1594,"date":"2021-07-22T16:38:56","date_gmt":"2021-07-22T16:38:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1594"},"modified":"2022-03-19T04:12:26","modified_gmt":"2022-03-19T04:12:26","slug":"helpful-integration-tools","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/helpful-integration-tools\/","title":{"raw":"Helpful Integration Tools","rendered":"Helpful Integration Tools"},"content":{"raw":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use a table of integrals to solve integration problems<\/li>\r\n \t<li>Use a computer algebra system (CAS) to solve integration problems<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section id=\"fs-id1165040774511\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Tables of Integrals<\/h2>\r\n<p id=\"fs-id1165042105278\">Integration tables, if used in the right manner, can be a handy way either to evaluate or check an integral quickly. Keep in mind that when using a table to check an answer, it is possible for two completely correct solutions to look very different. For example, in Trigonometric Substitution, we found that, by using the substitution [latex]x=\\tan\\theta [\/latex], we can arrive at<\/p>\r\n\r\n<div id=\"fs-id1165040681970\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+{x}^{2}}}=\\text{ln}\\left(x+\\sqrt{{x}^{2}+1}\\right)+C[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165041951596\">However, using [latex]x=\\text{sinh}\\theta [\/latex], we obtained a different solution\u2014namely,<\/p>\r\n\r\n<div id=\"fs-id1165041848129\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+{x}^{2}}}={\\text{sinh}}^{-1}x+C[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165041948411\">We later showed algebraically that the two solutions are equivalent. That is, we showed that [latex]{\\text{sinh}}^{-1}x=\\text{ln}\\left(x+\\sqrt{{x}^{2}+1}\\right)[\/latex]. In this case, the two antiderivatives that we found were actually equal. This need not be the case. However, as long as the difference in the two antiderivatives is a constant, they are equivalent.<\/p>\r\n\r\n<div id=\"fs-id1165041930726\" data-type=\"example\">\r\n<div id=\"fs-id1165041787566\" data-type=\"exercise\">\r\n<div id=\"fs-id1165042085684\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Using a Formula from a Table to Evaluate an Integral<\/h3>\r\n<div id=\"fs-id1165042085684\" data-type=\"problem\">\r\n<p id=\"fs-id1165041973418\">Use the table formula<\/p>\r\n\r\n<div id=\"fs-id1165041758432\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{a}^{2}-{u}^{2}}}{u}-{\\sin}^{-1}\\frac{u}{a}+C[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165040771022\">to evaluate [latex]\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{x}}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558899\"]\r\n<div id=\"fs-id1165041913720\" data-type=\"solution\">\r\n<p id=\"fs-id1165041766265\">If we look at integration tables, we see that several formulas contain expressions of the form [latex]\\sqrt{{a}^{2}-{u}^{2}}[\/latex]. This expression is actually similar to [latex]\\sqrt{16-{e}^{2x}}[\/latex], where [latex]a=4[\/latex] and [latex]u={e}^{x}[\/latex]. Keep in mind that we must also have [latex]du={e}^{x}[\/latex]. Multiplying the numerator and the denominator of the given integral by [latex]{e}^{x}[\/latex] should help to put this integral in a useful form. Thus, we now have<\/p>\r\n\r\n<div id=\"fs-id1165040752681\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{x}}dx=\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165041797345\">Substituting [latex]u={e}^{x}[\/latex] and [latex]du={e}^{x}[\/latex] produces [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du[\/latex]. From the integration table (#88 in Appendix A),<\/p>\r\n\r\n<div id=\"fs-id1165042045994\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{a}^{2}-{u}^{2}}}{u}-{\\sin}^{-1}\\frac{u}{a}+C[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165041845825\">Thus,<\/p>\r\n\r\n<div id=\"fs-id1165040798665\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccccc}\\hfill {\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{x}}dx}&amp; ={\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx}\\hfill &amp; &amp; &amp; \\text{Substitute}u={e}^{x}\\text{and}du={e}^{x}dx.\\hfill \\\\ &amp; ={\\displaystyle\\int \\frac{\\sqrt{{4}^{2}-{u}^{2}}}{{u}^{2}}du}\\hfill &amp; &amp; &amp; \\text{Apply the formula using}a=4.\\hfill \\\\ &amp; =-\\frac{\\sqrt{{4}^{2}-{u}^{2}}}{u}-{\\sin}^{-1}\\frac{u}{4}+C\\hfill &amp; &amp; &amp; \\text{Substitute}u={e}^{x}.\\hfill \\\\ &amp; =-\\frac{\\sqrt{16-{e}^{2x}}}{u}-{\\sin}^{-1}\\left(\\frac{{e}^{x}}{4}\\right)+C.\\hfill &amp; &amp; &amp; \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example:\u00a0Using a Formula from a Table to Evaluate an Integral[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6722645&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=CE2ryAVDgJ4&amp;video_target=tpm-plugin-248rwand-CE2ryAVDgJ4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.5.1_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \"3.5.1\" here (opens in new window)<\/a>.\r\n\r\n<\/section><section id=\"fs-id1165041803619\" data-depth=\"1\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]169275[\/ohm_question]\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Computer Algebra Systems<\/h2>\r\n<p id=\"fs-id1165042276048\">If available, a CAS is a faster alternative to a table for solving an integration problem. Many such systems are widely available and are, in general, quite easy to use.<\/p>\r\n\r\n<div id=\"fs-id1165040682585\" data-type=\"example\">\r\n<div id=\"fs-id1165041917465\" data-type=\"exercise\">\r\n<div id=\"fs-id1165041813674\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Using a Computer Algebra System to Evaluate an Integral<\/h3>\r\n<div id=\"fs-id1165041813674\" data-type=\"problem\">\r\n<p id=\"fs-id1165041762452\">Use a computer algebra system to evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}[\/latex]. Compare this result with [latex]\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}}{2}+\\frac{x}{2}|+C[\/latex], a result we might have obtained if we had used trigonometric substitution.<\/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-id1165041813679\" data-type=\"solution\">\r\n<p id=\"fs-id1165041826788\">Using Wolfram Alpha, we obtain<\/p>\r\n\r\n<div id=\"fs-id1165042228130\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}=\\text{ln}|\\sqrt{{x}^{2}-4}+x|+C[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042232336\">Notice that<\/p>\r\n\r\n<div id=\"fs-id1165042122802\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}}{2}+\\frac{x}{2}|+C=\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}+x}{2}|+C=\\text{ln}|\\sqrt{{x}^{2}-4}+x|-\\text{ln}2+C[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165041923020\">Since these two antiderivatives differ by only a constant, the solutions are equivalent. We could have also demonstrated that each of these antiderivatives is correct by differentiating them.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165041798149\" class=\"media-2\" data-type=\"note\">\r\n<div class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1165040745248\">You can access <a href=\"https:\/\/www.integral-calculator.com\/\" target=\"_blank\" rel=\"noopener\">this integral calculator for more practice calculating integrals<\/a>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165042089942\" data-type=\"example\">\r\n<div id=\"fs-id1165041762163\" data-type=\"exercise\">\r\n<div id=\"fs-id1165041932784\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Using a CAS to Evaluate an Integral<\/h3>\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example:\u00a0Using a CAS to Evaluate an Integral[\/caption]\r\n\r\n<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\"><\/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<div id=\"fs-id1165042199491\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1165041893152\" data-type=\"exercise\">\r\n<div id=\"fs-id1165040797562\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use a table of integrals to solve integration problems<\/li>\n<li>Use a computer algebra system (CAS) to solve integration problems<\/li>\n<\/ul>\n<\/div>\n<section id=\"fs-id1165040774511\" data-depth=\"1\">\n<h2 data-type=\"title\">Tables of Integrals<\/h2>\n<p id=\"fs-id1165042105278\">Integration tables, if used in the right manner, can be a handy way either to evaluate or check an integral quickly. Keep in mind that when using a table to check an answer, it is possible for two completely correct solutions to look very different. For example, in Trigonometric Substitution, we found that, by using the substitution [latex]x=\\tan\\theta[\/latex], we can arrive at<\/p>\n<div id=\"fs-id1165040681970\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+{x}^{2}}}=\\text{ln}\\left(x+\\sqrt{{x}^{2}+1}\\right)+C[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165041951596\">However, using [latex]x=\\text{sinh}\\theta[\/latex], we obtained a different solution\u2014namely,<\/p>\n<div id=\"fs-id1165041848129\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+{x}^{2}}}={\\text{sinh}}^{-1}x+C[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165041948411\">We later showed algebraically that the two solutions are equivalent. That is, we showed that [latex]{\\text{sinh}}^{-1}x=\\text{ln}\\left(x+\\sqrt{{x}^{2}+1}\\right)[\/latex]. In this case, the two antiderivatives that we found were actually equal. This need not be the case. However, as long as the difference in the two antiderivatives is a constant, they are equivalent.<\/p>\n<div id=\"fs-id1165041930726\" data-type=\"example\">\n<div id=\"fs-id1165041787566\" data-type=\"exercise\">\n<div id=\"fs-id1165042085684\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Using a Formula from a Table to Evaluate an Integral<\/h3>\n<div id=\"fs-id1165042085684\" data-type=\"problem\">\n<p id=\"fs-id1165041973418\">Use the table formula<\/p>\n<div id=\"fs-id1165041758432\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{a}^{2}-{u}^{2}}}{u}-{\\sin}^{-1}\\frac{u}{a}+C[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165040771022\">to evaluate [latex]\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{x}}dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558899\">Show Solution<\/span><\/p>\n<div id=\"q44558899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041913720\" data-type=\"solution\">\n<p id=\"fs-id1165041766265\">If we look at integration tables, we see that several formulas contain expressions of the form [latex]\\sqrt{{a}^{2}-{u}^{2}}[\/latex]. This expression is actually similar to [latex]\\sqrt{16-{e}^{2x}}[\/latex], where [latex]a=4[\/latex] and [latex]u={e}^{x}[\/latex]. Keep in mind that we must also have [latex]du={e}^{x}[\/latex]. Multiplying the numerator and the denominator of the given integral by [latex]{e}^{x}[\/latex] should help to put this integral in a useful form. Thus, we now have<\/p>\n<div id=\"fs-id1165040752681\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{x}}dx=\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165041797345\">Substituting [latex]u={e}^{x}[\/latex] and [latex]du={e}^{x}[\/latex] produces [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du[\/latex]. From the integration table (#88 in Appendix A),<\/p>\n<div id=\"fs-id1165042045994\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{a}^{2}-{u}^{2}}}{u}-{\\sin}^{-1}\\frac{u}{a}+C[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165041845825\">Thus,<\/p>\n<div id=\"fs-id1165040798665\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccccc}\\hfill {\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{x}}dx}& ={\\displaystyle\\int \\frac{\\sqrt{16-{e}^{2x}}}{{e}^{2x}}{e}^{x}dx}\\hfill & & & \\text{Substitute}u={e}^{x}\\text{and}du={e}^{x}dx.\\hfill \\\\ & ={\\displaystyle\\int \\frac{\\sqrt{{4}^{2}-{u}^{2}}}{{u}^{2}}du}\\hfill & & & \\text{Apply the formula using}a=4.\\hfill \\\\ & =-\\frac{\\sqrt{{4}^{2}-{u}^{2}}}{u}-{\\sin}^{-1}\\frac{u}{4}+C\\hfill & & & \\text{Substitute}u={e}^{x}.\\hfill \\\\ & =-\\frac{\\sqrt{16-{e}^{2x}}}{u}-{\\sin}^{-1}\\left(\\frac{{e}^{x}}{4}\\right)+C.\\hfill & & & \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example:\u00a0Using a Formula from a Table to Evaluate an Integral<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6722645&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=CE2ryAVDgJ4&amp;video_target=tpm-plugin-248rwand-CE2ryAVDgJ4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.5.1_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;3.5.1&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<section id=\"fs-id1165041803619\" data-depth=\"1\">\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm169275\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=169275&theme=oea&iframe_resize_id=ohm169275&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2 data-type=\"title\">Computer Algebra Systems<\/h2>\n<p id=\"fs-id1165042276048\">If available, a CAS is a faster alternative to a table for solving an integration problem. Many such systems are widely available and are, in general, quite easy to use.<\/p>\n<div id=\"fs-id1165040682585\" data-type=\"example\">\n<div id=\"fs-id1165041917465\" data-type=\"exercise\">\n<div id=\"fs-id1165041813674\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Using a Computer Algebra System to Evaluate an Integral<\/h3>\n<div id=\"fs-id1165041813674\" data-type=\"problem\">\n<p id=\"fs-id1165041762452\">Use a computer algebra system to evaluate [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}[\/latex]. Compare this result with [latex]\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}}{2}+\\frac{x}{2}|+C[\/latex], a result we might have obtained if we had used trigonometric substitution.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558898\">Show Solution<\/span><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041813679\" data-type=\"solution\">\n<p id=\"fs-id1165041826788\">Using Wolfram Alpha, we obtain<\/p>\n<div id=\"fs-id1165042228130\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-4}}=\\text{ln}|\\sqrt{{x}^{2}-4}+x|+C[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042232336\">Notice that<\/p>\n<div id=\"fs-id1165042122802\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}}{2}+\\frac{x}{2}|+C=\\text{ln}|\\frac{\\sqrt{{x}^{2}-4}+x}{2}|+C=\\text{ln}|\\sqrt{{x}^{2}-4}+x|-\\text{ln}2+C[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165041923020\">Since these two antiderivatives differ by only a constant, the solutions are equivalent. We could have also demonstrated that each of these antiderivatives is correct by differentiating them.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165041798149\" class=\"media-2\" data-type=\"note\">\n<div class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1165040745248\">You can access <a href=\"https:\/\/www.integral-calculator.com\/\" target=\"_blank\" rel=\"noopener\">this integral calculator for more practice calculating integrals<\/a>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165042089942\" data-type=\"example\">\n<div id=\"fs-id1165041762163\" data-type=\"exercise\">\n<div id=\"fs-id1165041932784\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Using a CAS to Evaluate an Integral<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558897\">Show Solution<\/span><\/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><div style=\"width: 504px\" 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\" \/><\/p>\n<p 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.<\/p>\n<\/div>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example:\u00a0Using a CAS to Evaluate an Integral<\/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\"><\/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<div id=\"fs-id1165042199491\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1165041893152\" data-type=\"exercise\">\n<div id=\"fs-id1165040797562\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558895\">Hint<\/span><\/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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558896\">Show Solution<\/span><\/p>\n<div id=\"q44558896\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1594\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>3.5.1. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>3.5.2. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"3.5.1\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"3.5.2\",\"author\":\"Ryan 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