{"id":778,"date":"2025-06-20T17:12:08","date_gmt":"2025-06-20T17:12:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=778"},"modified":"2025-09-05T17:54:15","modified_gmt":"2025-09-05T17:54:15","slug":"improper-integrals-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/improper-integrals-fresh-take\/","title":{"raw":"Improper Integrals: Fresh Take","rendered":"Improper Integrals: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate integrals over infinite intervals<\/li>\r\n \t<li>Find integrals when there's an infinite discontinuity inside your interval<\/li>\r\n \t<li>Use the comparison theorem to determine if an improper integral converges<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Integrating over an Infinite Interval<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Improper integrals handle two situations that break the rules of regular integration: infinite limits of integration and functions with discontinuities. We solve these by replacing the problematic parts with limits, transforming impossible calculations into manageable ones.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Three Types of Infinite Limits:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Type 1<\/strong>: [latex]\\int_a^{+\\infty} f(x) , dx = \\lim_{t \\to +\\infty} \\int_a^t f(x) , dx[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Type 2<\/strong>: [latex]\\int_{-\\infty}^b f(x) , dx = \\lim_{t \\to -\\infty} \\int_t^b f(x) , dx[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Type 3<\/strong>: [latex]\\int_{-\\infty}^{+\\infty} f(x) , dx = \\int_{-\\infty}^c f(x) , dx + \\int_c^{+\\infty} f(x) , dx[\/latex] (split at any point [latex]c[\/latex])<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Convergence vs. Divergence:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Converges<\/strong>: The limit exists and equals a finite number<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Diverges<\/strong>: The limit doesn't exist or approaches [latex]\\pm\\infty[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">For Type 3, both pieces must converge for the whole integral to converge.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem -Solving Strategy:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Set up the limit<\/strong> by replacing infinity with a variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Integrate normally<\/strong> over the finite interval<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Evaluate the limit<\/strong> as the variable approaches infinity<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Use L'H\u00f4pital's rule<\/strong> if needed for indeterminate forms<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165042988255\" data-type=\"problem\">\r\n<p id=\"fs-id1165043373560\">Evaluate [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{1}{{x}^{2}+4}dx[\/latex]. State whether the improper integral converges or diverges.<\/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-id1165043097500\" data-type=\"solution\">\r\n<p id=\"fs-id1165042534998\">Begin by rewriting [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{1}{{x}^{2}+4}dx[\/latex] as a limit using the equation 2 from the definition. Thus,<\/p>\r\n\r\n<div id=\"fs-id1165042316105\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccccc}\\hfill {\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{1}{{x}^{2}+4}dx&amp; =\\underset{x\\to \\text{-}\\infty }{\\text{lim}}{\\displaystyle\\int }_{t}^{0}\\frac{1}{{x}^{2}+4}dx\\hfill &amp; &amp; &amp; \\text{Rewrite as a limit.}\\hfill \\\\ &amp; =\\underset{t\\to \\text{-}\\infty }{\\text{lim}}\\frac{1}{2}{\\tan}^{-1}\\frac{x}{2}|{}_{\\begin{array}{c}\\\\ t\\end{array}}^{\\begin{array}{c}0\\\\ \\end{array}}\\hfill &amp; &amp; &amp; \\text{Find the antiderivative.}\\hfill \\\\ &amp; =\\frac{1}{2}\\underset{t\\to \\text{-}\\infty }{\\text{lim}}\\left({\\tan}^{-1}0-{\\tan}^{-1}\\frac{t}{2}\\right)\\hfill &amp; &amp; &amp; \\text{Evaluate the antiderivative.}\\hfill \\\\ &amp; =\\frac{\\pi }{4}.\\hfill &amp; &amp; &amp; \\text{Evaluate the limit and simplify.}\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165043422657\">The improper integral converges to [latex]\\frac{\\pi }{4}[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165043103020\" data-type=\"problem\">\r\n<p id=\"fs-id1165043103022\">Evaluate [latex]{\\displaystyle\\int }_{-3}^{+\\infty }{e}^{\\text{-}x}dx[\/latex]. State whether the improper integral converges or diverges.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558894\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1165043272721\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165042706105\">[latex]{\\displaystyle\\int }_{-3}^{+\\infty }{e}^{\\text{-}x}dx=\\underset{t\\to \\text{+}\\infty }{\\text{lim}}{\\displaystyle\\int }_{-3}^{t}{e}^{\\text{-}x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1165042355763\" data-type=\"solution\">\r\n<p id=\"fs-id1165042355765\">[latex]{e}^{3}[\/latex], converges<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8QzQO748O58?controls=0&amp;start=676&amp;end=750&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.7ImproperIntegrals676to750_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.7 Improper Integrals\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Integrating a Discontinuous Integrand<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Sometimes the problem isn't an infinite interval\u2014it's that your function has a vertical asymptote or other discontinuity right in the middle of where you want to integrate. These improper integrals require the same limit approach, but now you're approaching the troublesome point from one or both sides.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Three Discontinuity Cases:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Type 1 - Right Endpoint Problem<\/strong>: [latex]\\int_a^b f(x) , dx = \\lim_{t \\to b^-} \\int_a^t f(x) , dx[\/latex] (Function continuous on [latex][a,b)[\/latex], discontinuous at [latex]b[\/latex])<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Type 2 - Left Endpoint Problem<\/strong>: [latex]\\int_a^b f(x) , dx = \\lim_{t \\to a^+} \\int_t^b f(x) , dx[\/latex] (Function continuous on [latex](a,b][\/latex], discontinuous at [latex]a[\/latex])<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Type 3 - Interior Discontinuity<\/strong>: [latex]\\int_a^b f(x) , dx = \\int_a^c f(x) , dx + \\int_c^b f(x) , dx[\/latex] (Function has discontinuity at some point [latex]c[\/latex] inside [latex][a,b][\/latex])<\/p>\r\n<p class=\"whitespace-normal break-words\">Critical Rule for Type 3: Both pieces must converge for the whole integral to converge. If either piece diverges, the entire integral diverges.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem -Solving Strategy:<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify the discontinuity<\/strong> - where does the function blow up or have a jump?<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Set up the appropriate limit<\/strong> based on where the discontinuity occurs<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Integrate normally<\/strong> over the continuous parts<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Evaluate the limit<\/strong> as you approach the problematic point<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Use L'H\u00f4pital's rule<\/strong> if you encounter indeterminate forms<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165042376194\" data-type=\"problem\">\r\n<p id=\"fs-id1165042376196\">Evaluate [latex]{\\displaystyle\\int }_{0}^{2}\\frac{1}{x}dx[\/latex]. State whether the integral converges or diverges.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558889\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558889\"]\r\n<div id=\"fs-id1165042422655\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165042708426\">Write [latex]{\\displaystyle\\int }_{0}^{2}\\frac{1}{x}dx[\/latex] in limit form using equation 2 from the definition.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558890\"]\r\n<div id=\"fs-id1165042422640\" data-type=\"solution\">\r\n<p id=\"fs-id1165042422642\">[latex]+\\infty [\/latex], diverges<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8QzQO748O58?controls=0&amp;start=1388&amp;end=1468&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.7ImproperIntegrals1388to1468_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.7 Improper Integrals\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h2>A Comparison Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">When you can't evaluate an improper integral directly, the comparison test lets you determine convergence or divergence by comparing your function to a simpler one whose behavior you already understand. It's like using a known yardstick to measure an unknown quantity.<\/p>\r\n<p class=\"whitespace-normal break-words\">If [latex]0 \\leq f(x) \\leq g(x)[\/latex] for [latex]x \\geq a[\/latex], then:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Larger function dominates<\/strong>: If the smaller function [latex]f(x)[\/latex] diverges, then the larger function [latex]g(x)[\/latex] must also diverge<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Smaller function is bounded<\/strong>: If the larger function [latex]g(x)[\/latex] converges, then the smaller function [latex]f(x)[\/latex] must also converge<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>The Two Comparison Cases:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Case 1 - Proving Divergence<\/strong>: Find a smaller function that you know diverges If [latex]\\int_a^{+\\infty} f(x) , dx = +\\infty[\/latex] and [latex]f(x) \\leq g(x)[\/latex], then [latex]\\int_a^{+\\infty} g(x) , dx = +\\infty[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Case 2 - Proving Convergence<\/strong>: Find a larger function that you know converges\r\nIf [latex]\\int_a^{+\\infty} g(x) , dx[\/latex] converges and [latex]f(x) \\leq g(x)[\/latex], then [latex]\\int_a^{+\\infty} f(x) , dx[\/latex] converges<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Helpful Tips:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Common comparison functions<\/strong>: [latex]\\frac{1}{x^p}[\/latex] (diverges for [latex]p \\leq 1[\/latex], converges for [latex]p &gt; 1[\/latex]), [latex]e^{-x}[\/latex] (converges)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Reciprocal rule<\/strong>: If [latex]0 &lt; f(x) \\leq g(x)[\/latex], then [latex]\\frac{1}{f(x)} \\geq \\frac{1}{g(x)}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Focus on the tail<\/strong>: For large [latex]x[\/latex], identify the dominant behavior of your function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use a comparison to show that [latex]{\\displaystyle\\int }_{e}^{+\\infty }\\frac{\\text{ln}x}{x}dx[\/latex] diverges.\r\n[reveal-answer q=\"907753222\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"907753222\"]\r\n[latex]\\frac{1}{x}\\le \\frac{\\text{ln}x}{x}[\/latex] on [latex]\\left[e,\\text{+}\\infty \\right)[\/latex]\r\n[\/hidden-answer][reveal-answer q=\"6657883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"6657883\"]\r\nSince [latex]{\\displaystyle\\int }_{e}^{+\\infty }\\frac{1}{x}dx=\\text{+}\\infty [\/latex], [latex]{\\displaystyle\\int }_{e}^{+\\infty }\\frac{\\text{ln}x}{x}dx[\/latex] diverges\r\n[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8QzQO748O58?controls=0&amp;start=1783&amp;end=1837&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.7ImproperIntegrals1783to1837_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.7 Improper Integrals\" here (opens in new window)<\/a>.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate integrals over infinite intervals<\/li>\n<li>Find integrals when there&#8217;s an infinite discontinuity inside your interval<\/li>\n<li>Use the comparison theorem to determine if an improper integral converges<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Integrating over an Infinite Interval<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Improper integrals handle two situations that break the rules of regular integration: infinite limits of integration and functions with discontinuities. We solve these by replacing the problematic parts with limits, transforming impossible calculations into manageable ones.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Three Types of Infinite Limits:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Type 1<\/strong>: [latex]\\int_a^{+\\infty} f(x) , dx = \\lim_{t \\to +\\infty} \\int_a^t f(x) , dx[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Type 2<\/strong>: [latex]\\int_{-\\infty}^b f(x) , dx = \\lim_{t \\to -\\infty} \\int_t^b f(x) , dx[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Type 3<\/strong>: [latex]\\int_{-\\infty}^{+\\infty} f(x) , dx = \\int_{-\\infty}^c f(x) , dx + \\int_c^{+\\infty} f(x) , dx[\/latex] (split at any point [latex]c[\/latex])<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Convergence vs. Divergence:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Converges<\/strong>: The limit exists and equals a finite number<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Diverges<\/strong>: The limit doesn&#8217;t exist or approaches [latex]\\pm\\infty[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">For Type 3, both pieces must converge for the whole integral to converge.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem -Solving Strategy:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Set up the limit<\/strong> by replacing infinity with a variable<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Integrate normally<\/strong> over the finite interval<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Evaluate the limit<\/strong> as the variable approaches infinity<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Use L&#8217;H\u00f4pital&#8217;s rule<\/strong> if needed for indeterminate forms<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165042988255\" data-type=\"problem\">\n<p id=\"fs-id1165043373560\">Evaluate [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{1}{{x}^{2}+4}dx[\/latex]. State whether the improper integral converges or diverges.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Show Solution<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043097500\" data-type=\"solution\">\n<p id=\"fs-id1165042534998\">Begin by rewriting [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{1}{{x}^{2}+4}dx[\/latex] as a limit using the equation 2 from the definition. Thus,<\/p>\n<div id=\"fs-id1165042316105\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccccc}\\hfill {\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{1}{{x}^{2}+4}dx& =\\underset{x\\to \\text{-}\\infty }{\\text{lim}}{\\displaystyle\\int }_{t}^{0}\\frac{1}{{x}^{2}+4}dx\\hfill & & & \\text{Rewrite as a limit.}\\hfill \\\\ & =\\underset{t\\to \\text{-}\\infty }{\\text{lim}}\\frac{1}{2}{\\tan}^{-1}\\frac{x}{2}|{}_{\\begin{array}{c}\\\\ t\\end{array}}^{\\begin{array}{c}0\\\\ \\end{array}}\\hfill & & & \\text{Find the antiderivative.}\\hfill \\\\ & =\\frac{1}{2}\\underset{t\\to \\text{-}\\infty }{\\text{lim}}\\left({\\tan}^{-1}0-{\\tan}^{-1}\\frac{t}{2}\\right)\\hfill & & & \\text{Evaluate the antiderivative.}\\hfill \\\\ & =\\frac{\\pi }{4}.\\hfill & & & \\text{Evaluate the limit and simplify.}\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165043422657\">The improper integral converges to [latex]\\frac{\\pi }{4}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165043103020\" data-type=\"problem\">\n<p id=\"fs-id1165043103022\">Evaluate [latex]{\\displaystyle\\int }_{-3}^{+\\infty }{e}^{\\text{-}x}dx[\/latex]. State whether the improper integral converges or diverges.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558894\">Hint<\/button><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165043272721\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165042706105\">[latex]{\\displaystyle\\int }_{-3}^{+\\infty }{e}^{\\text{-}x}dx=\\underset{t\\to \\text{+}\\infty }{\\text{lim}}{\\displaystyle\\int }_{-3}^{t}{e}^{\\text{-}x}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558895\">Show Solution<\/button><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042355763\" data-type=\"solution\">\n<p id=\"fs-id1165042355765\">[latex]{e}^{3}[\/latex], converges<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8QzQO748O58?controls=0&amp;start=676&amp;end=750&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.7ImproperIntegrals676to750_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.7 Improper Integrals&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h2 data-type=\"title\">Integrating a Discontinuous Integrand<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Sometimes the problem isn&#8217;t an infinite interval\u2014it&#8217;s that your function has a vertical asymptote or other discontinuity right in the middle of where you want to integrate. These improper integrals require the same limit approach, but now you&#8217;re approaching the troublesome point from one or both sides.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Three Discontinuity Cases:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Type 1 &#8211; Right Endpoint Problem<\/strong>: [latex]\\int_a^b f(x) , dx = \\lim_{t \\to b^-} \\int_a^t f(x) , dx[\/latex] (Function continuous on [latex][a,b)[\/latex], discontinuous at [latex]b[\/latex])<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Type 2 &#8211; Left Endpoint Problem<\/strong>: [latex]\\int_a^b f(x) , dx = \\lim_{t \\to a^+} \\int_t^b f(x) , dx[\/latex] (Function continuous on [latex](a,b][\/latex], discontinuous at [latex]a[\/latex])<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Type 3 &#8211; Interior Discontinuity<\/strong>: [latex]\\int_a^b f(x) , dx = \\int_a^c f(x) , dx + \\int_c^b f(x) , dx[\/latex] (Function has discontinuity at some point [latex]c[\/latex] inside [latex][a,b][\/latex])<\/p>\n<p class=\"whitespace-normal break-words\">Critical Rule for Type 3: Both pieces must converge for the whole integral to converge. If either piece diverges, the entire integral diverges.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem -Solving Strategy:<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Identify the discontinuity<\/strong> &#8211; where does the function blow up or have a jump?<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Set up the appropriate limit<\/strong> based on where the discontinuity occurs<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Integrate normally<\/strong> over the continuous parts<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Evaluate the limit<\/strong> as you approach the problematic point<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Use L&#8217;H\u00f4pital&#8217;s rule<\/strong> if you encounter indeterminate forms<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165042376194\" data-type=\"problem\">\n<p id=\"fs-id1165042376196\">Evaluate [latex]{\\displaystyle\\int }_{0}^{2}\\frac{1}{x}dx[\/latex]. State whether the integral converges or diverges.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558889\">Hint<\/button><\/p>\n<div id=\"q44558889\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042422655\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165042708426\">Write [latex]{\\displaystyle\\int }_{0}^{2}\\frac{1}{x}dx[\/latex] in limit form using equation 2 from the definition.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558890\">Show Solution<\/button><\/p>\n<div id=\"q44558890\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042422640\" data-type=\"solution\">\n<p id=\"fs-id1165042422642\">[latex]+\\infty[\/latex], diverges<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8QzQO748O58?controls=0&amp;start=1388&amp;end=1468&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.7ImproperIntegrals1388to1468_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.7 Improper Integrals&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h2>A Comparison Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When you can&#8217;t evaluate an improper integral directly, the comparison test lets you determine convergence or divergence by comparing your function to a simpler one whose behavior you already understand. It&#8217;s like using a known yardstick to measure an unknown quantity.<\/p>\n<p class=\"whitespace-normal break-words\">If [latex]0 \\leq f(x) \\leq g(x)[\/latex] for [latex]x \\geq a[\/latex], then:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Larger function dominates<\/strong>: If the smaller function [latex]f(x)[\/latex] diverges, then the larger function [latex]g(x)[\/latex] must also diverge<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Smaller function is bounded<\/strong>: If the larger function [latex]g(x)[\/latex] converges, then the smaller function [latex]f(x)[\/latex] must also converge<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>The Two Comparison Cases:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Case 1 &#8211; Proving Divergence<\/strong>: Find a smaller function that you know diverges If [latex]\\int_a^{+\\infty} f(x) , dx = +\\infty[\/latex] and [latex]f(x) \\leq g(x)[\/latex], then [latex]\\int_a^{+\\infty} g(x) , dx = +\\infty[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Case 2 &#8211; Proving Convergence<\/strong>: Find a larger function that you know converges<br \/>\nIf [latex]\\int_a^{+\\infty} g(x) , dx[\/latex] converges and [latex]f(x) \\leq g(x)[\/latex], then [latex]\\int_a^{+\\infty} f(x) , dx[\/latex] converges<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Helpful Tips:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Common comparison functions<\/strong>: [latex]\\frac{1}{x^p}[\/latex] (diverges for [latex]p \\leq 1[\/latex], converges for [latex]p > 1[\/latex]), [latex]e^{-x}[\/latex] (converges)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Reciprocal rule<\/strong>: If [latex]0 < f(x) \\leq g(x)[\/latex], then [latex]\\frac{1}{f(x)} \\geq \\frac{1}{g(x)}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Focus on the tail<\/strong>: For large [latex]x[\/latex], identify the dominant behavior of your function<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use a comparison to show that [latex]{\\displaystyle\\int }_{e}^{+\\infty }\\frac{\\text{ln}x}{x}dx[\/latex] diverges.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q907753222\">Hint<\/button><\/p>\n<div id=\"q907753222\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\frac{1}{x}\\le \\frac{\\text{ln}x}{x}[\/latex] on [latex]\\left[e,\\text{+}\\infty \\right)[\/latex]\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q6657883\">Show Solution<\/button><\/p>\n<div id=\"q6657883\" class=\"hidden-answer\" style=\"display: none\">\nSince [latex]{\\displaystyle\\int }_{e}^{+\\infty }\\frac{1}{x}dx=\\text{+}\\infty[\/latex], [latex]{\\displaystyle\\int }_{e}^{+\\infty }\\frac{\\text{ln}x}{x}dx[\/latex] diverges\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/8QzQO748O58?controls=0&amp;start=1783&amp;end=1837&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.7ImproperIntegrals1783to1837_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.7 Improper Integrals&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":667,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/778"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/778\/revisions"}],"predecessor-version":[{"id":2231,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/778\/revisions\/2231"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/667"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/778\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=778"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=778"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=778"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}