{"id":488,"date":"2025-02-13T19:45:24","date_gmt":"2025-02-13T19:45:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integrals-exponential-functions-and-logarithms-fresh-take\/"},"modified":"2025-02-13T19:45:24","modified_gmt":"2025-02-13T19:45:24","slug":"integrals-exponential-functions-and-logarithms-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integrals-exponential-functions-and-logarithms-fresh-take\/","title":{"raw":"Integrals, Exponential Functions, and Logarithms: Fresh Take","rendered":"Integrals, Exponential Functions, and Logarithms: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Understand the natural logarithm and the mathematical constant e using integrals<\/li>\n\t<li>Identify how to differentiate the natural logarithm function<\/li>\n\t<li>Perform integrations where the natural logarithm is involved<\/li>\n\t<li>Understand how to find derivatives and integrals of exponential functions<\/li>\n\t<li>Convert logarithmic and exponential expressions to base e forms<\/li>\n<\/ul>\n<\/section>\n<h2>The Natural Logarithm as an Integral<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Definition of Natural Logarithm: [latex]\\ln x = \\int_1^x \\frac{1}{t} dt[\/latex] for [latex]x &gt; 0[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Key Properties:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx} \\ln x = \\frac{1}{x}[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{u} du = \\ln |u| + C[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\ln 1 = 0[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\ln(ab) = \\ln a + \\ln b[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\ln(\\frac{a}{b}) = \\ln a - \\ln b[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\ln(a^r) = r \\ln a[\/latex] (for rational r)<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Graphical Interpretation:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Area under [latex]y = \\frac{1}{t}[\/latex] from [latex]1[\/latex] to [latex]x (x &gt; 1)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Negative area under [latex]y = \\frac{1}{t}[\/latex] from [latex]x[\/latex] to [latex]1[\/latex] ([latex]0 &lt; x &lt; 1[\/latex])<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Recognize logarithmic forms in integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">Use logarithm properties to simplify expressions<\/li>\n\t<li class=\"whitespace-normal break-words\">Apply chain rule for derivatives of logarithmic functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Use u-substitution for integrals involving [latex]\\frac{1}{u}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167794334473\">Calculate the following derivatives:<\/p>\n<ol id=\"fs-id1167793964608\" style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{2}[\/latex]<\/li>\n<\/ol>\n\n[reveal-answer q=\"fs-id1167793259644\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793259644\"]\n\n<ol id=\"fs-id1167793259644\" style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)=\\frac{4x+1}{2{x}^{2}+x}[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{2}=\\frac{6\\text{ln}({x}^{3})}{x}[\/latex]<\/li>\n<\/ol>\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\/6_9o-2mK1tk?controls=0&amp;start=0&amp;end=135&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\/6.7TryItProblems0to135_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"6.7 Try It Problems\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Calculate the integral [latex]\\displaystyle\\int \\frac{x}{{x}^{2}+4}dx.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793887223\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793887223\"]\n\n<p id=\"fs-id1167793887223\">Using [latex]u[\/latex]-substitution, let [latex]u={x}^{2}+4.[\/latex] Then [latex]du=2xdx[\/latex] and we have<\/p>\n<div id=\"fs-id1167793971678\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\frac{x}{{x}^{2}+4}dx=\\frac{1}{2}\\displaystyle\\int \\frac{1}{u}du\\frac{1}{2}\\text{ln}|u|+C=\\frac{1}{2}\\text{ln}|{x}^{2}+4|+C=\\frac{1}{2}\\text{ln}({x}^{2}+4)+C.[\/latex][\/hidden-answer]<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793607833\">Use properties of logarithms to simplify the following expression into a single logarithm:<\/p>\n<div id=\"fs-id1167793607837\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{ln}8-\\text{ln}2-\\text{ln}(\\frac{1}{4}).[\/latex]<\/div>\n\n[reveal-answer q=\"fs-id1167793949542\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793949542\"][latex]4\\text{ln}2[\/latex]\n\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Evaluate [latex]\\int \\frac{x^3 + 2x}{x^4 + 4x^2 + 4} dx[\/latex]<\/p>\n<p><br>\n[reveal-answer q=\"76034\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"76034\"]<\/p>\n<p>Recognize this as [latex]\\int \\frac{1}{u} \\cdot \\frac{du}{dx} dx[\/latex]&nbsp;<\/p>\n<p>Let [latex]u = x^4 + 4x^2 + 4 = (x^2 + 2)^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{du}{dx} = 4x^3 + 8x = 4x(x^2 + 2)[\/latex]<\/p>\n<p>Rewrite integral:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4} \\int \\frac{1}{(x^2 + 2)^2} \\cdot 4x(x^2 + 2) dx = \\frac{1}{4} \\int \\frac{1}{x^2 + 2} dx[\/latex]<\/p>\n<p>Final answer:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4} \\ln|x^2 + 2| + C[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h2>Properties of the Exponential Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Definition of [latex]e[\/latex]:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]e[\/latex] is the unique number such that [latex]\\ln e = 1[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Approximately [latex]2.71828182846...[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Exponential Function:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Defined as the inverse of [latex]\\ln x[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]e^x = \\exp(x)[\/latex] for all real [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Key Properties:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]e^{\\ln x} = x[\/latex] for [latex]x &gt; 0[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\ln(e^x) = x[\/latex] for all [latex]x[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]e^p e^q = e^{p+q}[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{e^p}{e^q} = e^{p-q}[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]{({e}^{p})}^{r}={e}^{pr}[\/latex] (r rational)<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Derivatives and Integrals:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}e^x = e^x[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]\\int e^x dx = e^x + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793374990\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167793374993\" style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}[\/latex]<\/li>\n<\/ol>\n<p>[reveal-answer q=\"fs-id1167793872351\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793872351\"]<\/p>\n<ol id=\"fs-id1167793872351\" style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})={e}^{{x}^{2}-5x}(2x-5)[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}=6{e}^{6t}[\/latex]<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6_9o-2mK1tk?controls=0&amp;start=238&amp;end=356&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\/6.7TryItProblems238to356_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"6.7 Try It Problems\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the following integral: [latex]\\displaystyle\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793501968\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793501968\"]\n\n<p id=\"fs-id1167793501968\">Using [latex]u[\/latex]-substitution, let [latex]u=\\text{\u2212}{x}^{2}.[\/latex] Then [latex]du=-2xdx,[\/latex] and we have<\/p>\n<div id=\"fs-id1167793956578\" class=\"equation unnumbered\">[latex]\\displaystyle\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx=\\text{\u2212}\\displaystyle\\int {e}^{u}du=\\text{\u2212}{e}^{u}+C=\\text{\u2212}{e}^{\\text{\u2212}{x}^{2}}+C.[\/latex][\/hidden-answer]<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find [latex]\\frac{d}{dx}(e^{2x^3 + \\sin x})[\/latex]<\/p>\n<p><br>\n[reveal-answer q=\"303335\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"303335\"]<\/p>\n<p>Recognize this as a composition of functions<\/p>\n<p>Apply the chain rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^{2x^3 + \\sin x}) = e^{2x^3 + \\sin x} \\cdot \\frac{d}{dx}(2x^3 + \\sin x)[\/latex]<\/p>\n<p>Evaluate the inner derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(2x^3 + \\sin x) = 6x^2 + \\cos x[\/latex]<\/p>\n<p>Combine:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^{2x^3 + \\sin x}) = e^{2x^3 + \\sin x}(6x^2 + \\cos x)[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h2>General Logarithmic and Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Definition of General Exponential Functions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For [latex]a &gt; 0[\/latex] and any real [latex]x[\/latex]: [latex]y = a^x = e^{x \\ln a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Derivatives of General Exponential Functions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx} a^x = a^x \\ln a[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Integrals of General Exponential Functions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\int a^x dx = \\frac{1}{\\ln a} a^x + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">General Logarithm Functions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Inverse of [latex]a^x[\/latex] when [latex]a \\neq 1[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]y = \\log_a x[\/latex] if and only if [latex]x = a^y[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Relationship to Natural Logarithm:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\log_a x = \\frac{\\ln x}{\\ln a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Derivatives of General Logarithm Functions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx} \\log_a x = \\frac{1}{x \\ln a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793543534\">Evaluate the following derivatives:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})[\/latex]<\/li>\n<\/ol>\n\n[reveal-answer q=\"fs-id1167793978416\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793978416\"]\n\n<ol style=\"list-style-type: lower-alpha;\">\n\t<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}={4}^{{t}^{4}}(\\text{ln}4)(4{t}^{3})[\/latex]<\/li>\n\t<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})=\\frac{x}{(\\text{ln}3)({x}^{2}+1)}[\/latex]<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6_9o-2mK1tk?controls=0&amp;start=461&amp;end=662&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\/6.7TryItProblems461to662_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"6.7 Try It Problems\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the following integral: [latex]\\displaystyle\\int \\frac{3}{{2}^{3x}}dx.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793956537\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793956537\"]\n\n<p id=\"fs-id1167793956537\">Use [latex]u\\text{-substitution}[\/latex] and let [latex]u=-3x.[\/latex] Then [latex]du=-3dx[\/latex] and we have<\/p>\n<div id=\"fs-id1167793929418\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\frac{3}{{2}^{3x}}dx=\\displaystyle\\int 3\u00b7{2}^{-3x}dx=\\text{\u2212}\\displaystyle\\int {2}^{u}du=-\\frac{1}{\\text{ln}2}{2}^{u}+C=-\\frac{1}{\\text{ln}2}{2}^{-3x}+C.[\/latex][\/hidden-answer]<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Evaluate the following derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx} (3^x \\cdot \\log_2(x^2 + 1))[\/latex]<\/p>\n<p>[reveal-answer q=\"57025\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"57025\"]<\/p>\n<p>Use the product rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx} (u \\cdot v) = u' \\cdot v + u \\cdot v'[\/latex]<\/p>\n<p>Let [latex]u = 3^x[\/latex] and [latex]v = \\log_2(x^2 + 1)[\/latex]<\/p>\n<p>Calculate [latex]u'[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]u' = 3^x \\ln 3[\/latex]<\/p>\n<p>Calculate [latex]v'[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]v' = \\frac{1}{(x^2 + 1) \\ln 2} \\cdot \\frac{d}{dx}(x^2 + 1) = \\frac{2x}{(x^2 + 1) \\ln 2}[\/latex]<\/p>\n<p>Apply the product rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br>\n\\frac{d}{dx} (3^x \\cdot \\log_2(x^2 + 1)) &amp;=&amp; 3^x \\ln 3 \\cdot \\log_2(x^2 + 1) + 3^x \\cdot \\frac{2x}{(x^2 + 1) \\ln 2} \\\\<br>\n&amp;=&amp; 3^x \\left(\\ln 3 \\cdot \\log_2(x^2 + 1) + \\frac{2x}{(x^2 + 1) \\ln 2}\\right)<br>\n\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the natural logarithm and the mathematical constant e using integrals<\/li>\n<li>Identify how to differentiate the natural logarithm function<\/li>\n<li>Perform integrations where the natural logarithm is involved<\/li>\n<li>Understand how to find derivatives and integrals of exponential functions<\/li>\n<li>Convert logarithmic and exponential expressions to base e forms<\/li>\n<\/ul>\n<\/section>\n<h2>The Natural Logarithm as an Integral<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Natural Logarithm: [latex]\\ln x = \\int_1^x \\frac{1}{t} dt[\/latex] for [latex]x > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Key Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx} \\ln x = \\frac{1}{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{u} du = \\ln |u| + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\ln 1 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\ln(ab) = \\ln a + \\ln b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\ln(\\frac{a}{b}) = \\ln a - \\ln b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\ln(a^r) = r \\ln a[\/latex] (for rational r)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Graphical Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Area under [latex]y = \\frac{1}{t}[\/latex] from [latex]1[\/latex] to [latex]x (x > 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Negative area under [latex]y = \\frac{1}{t}[\/latex] from [latex]x[\/latex] to [latex]1[\/latex] ([latex]0 < x < 1[\/latex])<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Recognize logarithmic forms in integrals<\/li>\n<li class=\"whitespace-normal break-words\">Use logarithm properties to simplify expressions<\/li>\n<li class=\"whitespace-normal break-words\">Apply chain rule for derivatives of logarithmic functions<\/li>\n<li class=\"whitespace-normal break-words\">Use u-substitution for integrals involving [latex]\\frac{1}{u}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167794334473\">Calculate the following derivatives:<\/p>\n<ol id=\"fs-id1167793964608\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793259644\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793259644\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793259644\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dx}\\text{ln}(2{x}^{2}+x)=\\frac{4x+1}{2{x}^{2}+x}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{ln}({x}^{3}))}^{2}=\\frac{6\\text{ln}({x}^{3})}{x}[\/latex]<\/li>\n<\/ol>\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\/6_9o-2mK1tk?controls=0&amp;start=0&amp;end=135&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\/6.7TryItProblems0to135_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;6.7 Try It Problems&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Calculate the integral [latex]\\displaystyle\\int \\frac{x}{{x}^{2}+4}dx.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793887223\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793887223\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793887223\">Using [latex]u[\/latex]-substitution, let [latex]u={x}^{2}+4.[\/latex] Then [latex]du=2xdx[\/latex] and we have<\/p>\n<div id=\"fs-id1167793971678\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\int \\frac{x}{{x}^{2}+4}dx=\\frac{1}{2}\\displaystyle\\int \\frac{1}{u}du\\frac{1}{2}\\text{ln}|u|+C=\\frac{1}{2}\\text{ln}|{x}^{2}+4|+C=\\frac{1}{2}\\text{ln}({x}^{2}+4)+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793607833\">Use properties of logarithms to simplify the following expression into a single logarithm:<\/p>\n<div id=\"fs-id1167793607837\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{ln}8-\\text{ln}2-\\text{ln}(\\frac{1}{4}).[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793949542\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793949542\" class=\"hidden-answer\" style=\"display: none\">[latex]4\\text{ln}2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Evaluate [latex]\\int \\frac{x^3 + 2x}{x^4 + 4x^2 + 4} dx[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q76034\">Show Answer<\/button><\/p>\n<div id=\"q76034\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recognize this as [latex]\\int \\frac{1}{u} \\cdot \\frac{du}{dx} dx[\/latex]&nbsp;<\/p>\n<p>Let [latex]u = x^4 + 4x^2 + 4 = (x^2 + 2)^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{du}{dx} = 4x^3 + 8x = 4x(x^2 + 2)[\/latex]<\/p>\n<p>Rewrite integral:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4} \\int \\frac{1}{(x^2 + 2)^2} \\cdot 4x(x^2 + 2) dx = \\frac{1}{4} \\int \\frac{1}{x^2 + 2} dx[\/latex]<\/p>\n<p>Final answer:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4} \\ln|x^2 + 2| + C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Properties of the Exponential Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of [latex]e[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]e[\/latex] is the unique number such that [latex]\\ln e = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Approximately [latex]2.71828182846...[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Exponential Function:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Defined as the inverse of [latex]\\ln x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e^x = \\exp(x)[\/latex] for all real [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Properties:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]e^{\\ln x} = x[\/latex] for [latex]x > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\ln(e^x) = x[\/latex] for all [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]e^p e^q = e^{p+q}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{e^p}{e^q} = e^{p-q}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{({e}^{p})}^{r}={e}^{pr}[\/latex] (r rational)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivatives and Integrals:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}e^x = e^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int e^x dx = e^x + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793374990\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167793374993\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})[\/latex]<\/li>\n<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793872351\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793872351\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793872351\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dx}(\\frac{{e}^{{x}^{2}}}{{e}^{5x}})={e}^{{x}^{2}-5x}(2x-5)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dt}{({e}^{2t})}^{3}=6{e}^{6t}[\/latex]<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6_9o-2mK1tk?controls=0&amp;start=238&amp;end=356&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\/6.7TryItProblems238to356_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;6.7 Try It Problems&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the following integral: [latex]\\displaystyle\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793501968\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793501968\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793501968\">Using [latex]u[\/latex]-substitution, let [latex]u=\\text{\u2212}{x}^{2}.[\/latex] Then [latex]du=-2xdx,[\/latex] and we have<\/p>\n<div id=\"fs-id1167793956578\" class=\"equation unnumbered\">[latex]\\displaystyle\\int 2x{e}^{\\text{\u2212}{x}^{2}}dx=\\text{\u2212}\\displaystyle\\int {e}^{u}du=\\text{\u2212}{e}^{u}+C=\\text{\u2212}{e}^{\\text{\u2212}{x}^{2}}+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find [latex]\\frac{d}{dx}(e^{2x^3 + \\sin x})[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q303335\">Show Answer<\/button><\/p>\n<div id=\"q303335\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recognize this as a composition of functions<\/p>\n<p>Apply the chain rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^{2x^3 + \\sin x}) = e^{2x^3 + \\sin x} \\cdot \\frac{d}{dx}(2x^3 + \\sin x)[\/latex]<\/p>\n<p>Evaluate the inner derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(2x^3 + \\sin x) = 6x^2 + \\cos x[\/latex]<\/p>\n<p>Combine:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx}(e^{2x^3 + \\sin x}) = e^{2x^3 + \\sin x}(6x^2 + \\cos x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>General Logarithmic and Exponential Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of General Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]a > 0[\/latex] and any real [latex]x[\/latex]: [latex]y = a^x = e^{x \\ln a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivatives of General Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx} a^x = a^x \\ln a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Integrals of General Exponential Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\int a^x dx = \\frac{1}{\\ln a} a^x + C[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Logarithm Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Inverse of [latex]a^x[\/latex] when [latex]a \\neq 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = \\log_a x[\/latex] if and only if [latex]x = a^y[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship to Natural Logarithm:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\log_a x = \\frac{\\ln x}{\\ln a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Derivatives of General Logarithm Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx} \\log_a x = \\frac{1}{x \\ln a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793543534\">Evaluate the following derivatives:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793978416\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793978416\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\frac{d}{dt}{4}^{{t}^{4}}={4}^{{t}^{4}}(\\text{ln}4)(4{t}^{3})[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{\\text{log}}_{3}(\\sqrt{{x}^{2}+1})=\\frac{x}{(\\text{ln}3)({x}^{2}+1)}[\/latex]<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution to the above Try It.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/6_9o-2mK1tk?controls=0&amp;start=461&amp;end=662&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\/6.7TryItProblems461to662_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;6.7 Try It Problems&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Evaluate the following integral: [latex]\\displaystyle\\int \\frac{3}{{2}^{3x}}dx.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793956537\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793956537\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793956537\">Use [latex]u\\text{-substitution}[\/latex] and let [latex]u=-3x.[\/latex] Then [latex]du=-3dx[\/latex] and we have<\/p>\n<div id=\"fs-id1167793929418\" class=\"equation unnumbered\">[latex]\\displaystyle\\int \\frac{3}{{2}^{3x}}dx=\\displaystyle\\int 3\u00b7{2}^{-3x}dx=\\text{\u2212}\\displaystyle\\int {2}^{u}du=-\\frac{1}{\\text{ln}2}{2}^{u}+C=-\\frac{1}{\\text{ln}2}{2}^{-3x}+C.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Evaluate the following derivative:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx} (3^x \\cdot \\log_2(x^2 + 1))[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57025\">Show Answer<\/button><\/p>\n<div id=\"q57025\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the product rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{d}{dx} (u \\cdot v) = u' \\cdot v + u \\cdot v'[\/latex]<\/p>\n<p>Let [latex]u = 3^x[\/latex] and [latex]v = \\log_2(x^2 + 1)[\/latex]<\/p>\n<p>Calculate [latex]u'[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]u' = 3^x \\ln 3[\/latex]<\/p>\n<p>Calculate [latex]v'[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]v' = \\frac{1}{(x^2 + 1) \\ln 2} \\cdot \\frac{d}{dx}(x^2 + 1) = \\frac{2x}{(x^2 + 1) \\ln 2}[\/latex]<\/p>\n<p>Apply the product rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl}<br \/> \\frac{d}{dx} (3^x \\cdot \\log_2(x^2 + 1)) &=& 3^x \\ln 3 \\cdot \\log_2(x^2 + 1) + 3^x \\cdot \\frac{2x}{(x^2 + 1) \\ln 2} \\\\<br \/> &=& 3^x \\left(\\ln 3 \\cdot \\log_2(x^2 + 1) + \\frac{2x}{(x^2 + 1) \\ln 2}\\right)<br \/> \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":479,"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\/488"}],"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\/488\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/479"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/488\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=488"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=488"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=488"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}