{"id":411,"date":"2025-02-13T19:44:42","date_gmt":"2025-02-13T19:44:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integrals-involving-exponential-and-logarithmic-functions-fresh-take\/"},"modified":"2025-02-13T19:44:42","modified_gmt":"2025-02-13T19:44:42","slug":"integrals-involving-exponential-and-logarithmic-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integrals-involving-exponential-and-logarithmic-functions-fresh-take\/","title":{"raw":"Integrals Involving Exponential and Logarithmic Functions: Fresh Take","rendered":"Integrals Involving Exponential and Logarithmic Functions: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Perform integrations on functions that include exponential terms<\/li>\n\t<li>Solve integrals that feature logarithmic functions<\/li>\n<\/ul>\n<\/section>\n<h2>Integrals of Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Exponential functions are their own derivatives and integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">Key integration formulas:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\int e^x dx = e^x + C[\/latex] [latex]\\int a^x dx = \\frac{a^x}{\\ln a} + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitution is often used for more complex exponential integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">Exponential functions are common in real-life applications, especially in growth and decay scenarios<\/li>\n\t<li class=\"whitespace-normal break-words\">Integration Process:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For simple exponentials, apply the formula directly<\/li>\n\t<li class=\"whitespace-normal break-words\">For complex expressions, use substitution with u as the exponent of [latex]e[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitution Tips:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">If only one [latex]e[\/latex] exists, choose its exponent as [latex]u[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">If multiple [latex]e[\/latex]'s exist, choose the more complicated function as [latex]u[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Common Mistakes:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Don't treat exponents on e like polynomial exponents<\/li>\n\t<li class=\"whitespace-normal break-words\">Be careful when both exponentials and polynomials are present<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the antiderivative of the function using substitution: [latex]{x}^{2}{e}^{-2{x}^{3}}.[\/latex]<\/p>\n<p>[reveal-answer q=\"20881654\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"20881654\"]<\/p>\n<p id=\"fs-id1170571622632\">Let [latex]u[\/latex] equal the exponent on [latex]e[\/latex].<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"fs-id1170572269592\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572269592\"]<\/p>\n<p id=\"fs-id1170572269592\">[latex]\\displaystyle\\int {x}^{2}{e}^{-2{x}^{3}}dx=-\\frac{1}{6}{e}^{-2{x}^{3}}+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/V9NlSl17duk?controls=0&amp;start=88&amp;end=170&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>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\/Calculus1+Videos\/5.6.1_88to170_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.6.1\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]{e}^{x}{(3{e}^{x}-2)}^{2}.[\/latex]<\/p>\n<p>[reveal-answer q=\"366490\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"366490\"]<\/p>\n<p>Let [latex]u=3{e}^{x}-2u=3{e}^{x}-2.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"6711088\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"6711088\"]<\/p>\n<p id=\"fs-id1170572209127\">[latex]\\displaystyle\\int {e}^{x}{(3{e}^{x}-2)}^{2}dx=\\frac{1}{9}{(3{e}^{x}-2)}^{3}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/V9NlSl17duk?controls=0&amp;start=257&amp;end=336&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>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\/Calculus1+Videos\/5.6.1_257to336_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.6.1\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the indefinite integral [latex]\\displaystyle\\int 2{x}^{3}{e}^{{x}^{4}}dx.[\/latex]<\/p>\n<p>[reveal-answer q=\"212787664\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"212787664\"]<\/p>\n<p id=\"fs-id1170572232621\">Let [latex]u={x}^{4}.[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"fs-id1170572242344\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572242344\"]<\/p>\n<p id=\"fs-id1170572242344\">[latex]\\displaystyle\\int 2{x}^{3}{e}^{{x}^{4}}dx=\\frac{1}{2}{e}^{{x}^{4}}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate [latex]{\\displaystyle\\int }_{0}^{2}{e}^{2x}dx.[\/latex]<\/p>\n<p>[reveal-answer q=\"3355576\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"3355576\"]<\/p>\n<p>Let [latex]u=2x[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"fs-id1170572274760\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572274760\"]<\/p>\n<p>[latex]\\frac{1}{2}{\\displaystyle\\int }_{0}^{4}{e}^{u}du=\\frac{1}{2}({e}^{4}-1)[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<h3><span style=\"font-size: 16px; font-weight: 400;\">Suppose a population of <\/span><span class=\"no-emphasis\" style=\"font-size: 16px; font-weight: 400;\">fruit flies<\/span><span style=\"font-size: 16px; font-weight: 400;\"> increases at a rate of [latex]g(t)=2{e}^{0.02t},[\/latex] in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?<\/span><\/h3>\n<p>[reveal-answer q=\"fs-id1170572183855\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572183855\"]<\/p>\n<p id=\"fs-id1170572183855\">Let [latex]G(t)[\/latex] represent the number of flies in the population at time [latex]t[\/latex]. Applying the net change theorem, we have<\/p>\n<div id=\"fs-id1170571712220\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\\\ G(10)\\hfill &amp; =G(0)+{\\int }_{0}^{10}2{e}^{0.02t}dt\\hfill \\\\ &amp; =100+{\\left[\\frac{2}{0.02}{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ &amp; =100+{\\left[100{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ &amp; =100+100{e}^{0.2}-100\\hfill \\\\ &amp; \\approx 122.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572338467\">There are [latex]122 [\/latex] flies in the population after [latex]10[\/latex] days.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Suppose the rate of growth of the fly population is given by [latex]g(t)={e}^{0.01t},[\/latex] and the initial fly population is 100 flies. How many flies are in the population after 15 days?<\/p>\n<p>[reveal-answer q=\"fs-id1170571653080\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170571653080\"]There are [latex]116[\/latex] flies.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the definite integral using substitution: [latex]{\\displaystyle\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx.[\/latex]<\/p>\n<p>[reveal-answer q=\"43778229\"]Hint[\/reveal-answer]<br>\n[hidden-answer a=\"43778229\"]<\/p>\n<p id=\"fs-id1170572554429\">Let [latex]u=4{x}^{-2}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<p>[reveal-answer q=\"fs-id1170572628419\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572628419\"]<\/p>\n<p id=\"fs-id1170572628419\">[latex]{\\displaystyle\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx=\\frac{1}{8}\\left[{e}^{4}-e\\right][\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h2>Integrals&nbsp;Involving Logarithmic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Integration of reciprocal functions leads to logarithms<\/li>\n\t<li class=\"whitespace-normal break-words\">Key integration formulas for logarithmic functions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\int x^{-1} dx = \\ln |x| + C[\/latex] [latex]\\int \\ln x dx = x\\ln x - x + C[\/latex] [latex]\\int \\log_a x dx = \\frac{x}{\\ln a}(\\ln x - 1) + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitution is often used for more complex logarithmic integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">Logarithmic integrals appear in various applications, including entropy and information theory<\/li>\n\t<li class=\"whitespace-normal break-words\">Integration Process:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For simple reciprocal functions, apply the formula directly<\/li>\n\t<li class=\"whitespace-normal break-words\">For complex expressions, use substitution or rewrite in terms of natural logarithms<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Substitution Tips:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Look for expressions that can be rewritten as [latex]u^(-1)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Be prepared to adjust du when necessary<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Common Challenges:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Remember the absolute value in the logarithm formula<\/li>\n\t<li class=\"whitespace-normal break-words\">Be careful with the domain of logarithmic functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Pay attention to the base of the logarithm<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]\\dfrac{1}{x+2}.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1170572480516\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572480516\"]\n\n<p id=\"fs-id1170572480516\">[latex]\\text{ln}|x+2|+C[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]{\\text{log}}_{3}x.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1170571660214\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170571660214\"]\n\n<p id=\"fs-id1170571660214\">[latex]\\frac{x}{\\text{ln}3}(\\text{ln}x-1)+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8mW2bG6HoPE?controls=0&amp;start=274&amp;end=342&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>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\/Calculus1+Videos\/5.6.2_274to342_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"5.6.2\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Perform integrations on functions that include exponential terms<\/li>\n<li>Solve integrals that feature logarithmic functions<\/li>\n<\/ul>\n<\/section>\n<h2>Integrals of Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Exponential functions are their own derivatives and integrals<\/li>\n<li class=\"whitespace-normal break-words\">Key integration formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int e^x dx = e^x + C[\/latex] [latex]\\int a^x dx = \\frac{a^x}{\\ln a} + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitution is often used for more complex exponential integrals<\/li>\n<li class=\"whitespace-normal break-words\">Exponential functions are common in real-life applications, especially in growth and decay scenarios<\/li>\n<li class=\"whitespace-normal break-words\">Integration Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For simple exponentials, apply the formula directly<\/li>\n<li class=\"whitespace-normal break-words\">For complex expressions, use substitution with u as the exponent of [latex]e[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitution Tips:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If only one [latex]e[\/latex] exists, choose its exponent as [latex]u[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If multiple [latex]e[\/latex]&#8216;s exist, choose the more complicated function as [latex]u[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Common Mistakes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Don&#8217;t treat exponents on e like polynomial exponents<\/li>\n<li class=\"whitespace-normal break-words\">Be careful when both exponentials and polynomials are present<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the antiderivative of the function using substitution: [latex]{x}^{2}{e}^{-2{x}^{3}}.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q20881654\">Hint<\/button><\/p>\n<div id=\"q20881654\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571622632\">Let [latex]u[\/latex] equal the exponent on [latex]e[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572269592\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572269592\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572269592\">[latex]\\displaystyle\\int {x}^{2}{e}^{-2{x}^{3}}dx=-\\frac{1}{6}{e}^{-2{x}^{3}}+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/V9NlSl17duk?controls=0&amp;start=88&amp;end=170&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>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\/Calculus1+Videos\/5.6.1_88to170_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.6.1&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]{e}^{x}{(3{e}^{x}-2)}^{2}.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q366490\">Hint<\/button><\/p>\n<div id=\"q366490\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]u=3{e}^{x}-2u=3{e}^{x}-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q6711088\">Show Solution<\/button><\/p>\n<div id=\"q6711088\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572209127\">[latex]\\displaystyle\\int {e}^{x}{(3{e}^{x}-2)}^{2}dx=\\frac{1}{9}{(3{e}^{x}-2)}^{3}[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/V9NlSl17duk?controls=0&amp;start=257&amp;end=336&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>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\/Calculus1+Videos\/5.6.1_257to336_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.6.1&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the indefinite integral [latex]\\displaystyle\\int 2{x}^{3}{e}^{{x}^{4}}dx.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q212787664\">Hint<\/button><\/p>\n<div id=\"q212787664\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572232621\">Let [latex]u={x}^{4}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572242344\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572242344\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572242344\">[latex]\\displaystyle\\int 2{x}^{3}{e}^{{x}^{4}}dx=\\frac{1}{2}{e}^{{x}^{4}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate [latex]{\\displaystyle\\int }_{0}^{2}{e}^{2x}dx.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q3355576\">Hint<\/button><\/p>\n<div id=\"q3355576\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]u=2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572274760\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572274760\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}{\\displaystyle\\int }_{0}^{4}{e}^{u}du=\\frac{1}{2}({e}^{4}-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<h3><span style=\"font-size: 16px; font-weight: 400;\">Suppose a population of <\/span><span class=\"no-emphasis\" style=\"font-size: 16px; font-weight: 400;\">fruit flies<\/span><span style=\"font-size: 16px; font-weight: 400;\"> increases at a rate of [latex]g(t)=2{e}^{0.02t},[\/latex] in flies per day. If the initial population of fruit flies is 100 flies, how many flies are in the population after 10 days?<\/span><\/h3>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572183855\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572183855\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572183855\">Let [latex]G(t)[\/latex] represent the number of flies in the population at time [latex]t[\/latex]. Applying the net change theorem, we have<\/p>\n<div id=\"fs-id1170571712220\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\\\ G(10)\\hfill & =G(0)+{\\int }_{0}^{10}2{e}^{0.02t}dt\\hfill \\\\ & =100+{\\left[\\frac{2}{0.02}{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ & =100+{\\left[100{e}^{0.02t}\\right]|}_{0}^{10}\\hfill \\\\ & =100+100{e}^{0.2}-100\\hfill \\\\ & \\approx 122.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572338467\">There are [latex]122[\/latex] flies in the population after [latex]10[\/latex] days.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Suppose the rate of growth of the fly population is given by [latex]g(t)={e}^{0.01t},[\/latex] and the initial fly population is 100 flies. How many flies are in the population after 15 days?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571653080\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571653080\" class=\"hidden-answer\" style=\"display: none\">There are [latex]116[\/latex] flies.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the definite integral using substitution: [latex]{\\displaystyle\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q43778229\">Hint<\/button><\/p>\n<div id=\"q43778229\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572554429\">Let [latex]u=4{x}^{-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572628419\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572628419\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572628419\">[latex]{\\displaystyle\\int }_{1}^{2}\\frac{1}{{x}^{3}}{e}^{4{x}^{-2}}dx=\\frac{1}{8}\\left[{e}^{4}-e\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Integrals&nbsp;Involving Logarithmic Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Integration of reciprocal functions leads to logarithms<\/li>\n<li class=\"whitespace-normal break-words\">Key integration formulas for logarithmic functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int x^{-1} dx = \\ln |x| + C[\/latex] [latex]\\int \\ln x dx = x\\ln x - x + C[\/latex] [latex]\\int \\log_a x dx = \\frac{x}{\\ln a}(\\ln x - 1) + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitution is often used for more complex logarithmic integrals<\/li>\n<li class=\"whitespace-normal break-words\">Logarithmic integrals appear in various applications, including entropy and information theory<\/li>\n<li class=\"whitespace-normal break-words\">Integration Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For simple reciprocal functions, apply the formula directly<\/li>\n<li class=\"whitespace-normal break-words\">For complex expressions, use substitution or rewrite in terms of natural logarithms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Substitution Tips:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Look for expressions that can be rewritten as [latex]u^(-1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Be prepared to adjust du when necessary<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Common Challenges:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Remember the absolute value in the logarithm formula<\/li>\n<li class=\"whitespace-normal break-words\">Be careful with the domain of logarithmic functions<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the base of the logarithm<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]\\dfrac{1}{x+2}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572480516\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572480516\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572480516\">[latex]\\text{ln}|x+2|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the antiderivative of [latex]{\\text{log}}_{3}x.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170571660214\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170571660214\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170571660214\">[latex]\\frac{x}{\\text{ln}3}(\\text{ln}x-1)+C[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8mW2bG6HoPE?controls=0&amp;start=274&amp;end=342&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>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\/Calculus1+Videos\/5.6.2_274to342_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;5.6.2&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":12,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":399,"module-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\/411"}],"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\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/411\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/399"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/411\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=411"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=411"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=411"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}