{"id":718,"date":"2025-06-20T17:07:21","date_gmt":"2025-06-20T17:07:21","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=718"},"modified":"2025-09-05T17:33:21","modified_gmt":"2025-09-05T17:33:21","slug":"partial-fractions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/partial-fractions-fresh-take\/","title":{"raw":"Partial Fractions: Fresh Take","rendered":"Partial Fractions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Break down and integrate rational functions using partial fractions<\/li>\r\n \t<li>Identify and work with simple linear factors in rational functions<\/li>\r\n \t<li>Handle repeated linear factors when using partial fractions<\/li>\r\n \t<li>Work with quadratic factors in rational functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>What Is Partial Fraction Decomposition?<\/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\">Think of partial fraction decomposition as taking apart a complicated fraction to reveal the simple pieces hiding inside. Instead of combining fractions like [latex]\\frac{1}{x+1} + \\frac{2}{x-2}[\/latex], you're doing the reverse\u2014starting with something messy like [latex]\\frac{3x}{x^2-x-2}[\/latex] and splitting it back into those manageable pieces.<\/p>\r\nYou can only use partial fractions when the degree of the numerator is <strong>less than<\/strong> the degree of the denominator. If it's not, you must use polynomial long division first to create a \"proper\" fraction.\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>Check the degree requirement<\/strong> - Is deg(numerator) &lt; deg(denominator)?<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>If not, use polynomial long division<\/strong> to get [latex]\\frac{P(x)}{Q(x)} = A(x) + \\frac{R(x)}{Q(x)}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Factor the denominator<\/strong> completely<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Set up partial fractions<\/strong> based on the factors<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Solve for the unknown constants<\/strong><\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Integrate each simple fraction<\/strong><\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">You can always verify your decomposition by finding a common denominator and checking that you get back to the original fraction.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165040743552\" data-type=\"problem\">\r\n<p id=\"fs-id1165042047432\">Evaluate [latex]\\displaystyle\\int \\frac{x - 3}{x+2}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558897\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1165042133890\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165041977487\">Use long division to obtain [latex]\\frac{x - 3}{x+2}=1-\\frac{5}{x+2}[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558898\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1165041985161\" data-type=\"solution\">\r\n<p id=\"fs-id1165042050319\">[latex]x - 5\\text{ln}|x+2|+C[\/latex]<\/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\/STtozLX2gbk?controls=0&amp;start=135&amp;end=182&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions135to182_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.4 Partial Fractions\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Nonrepeated Linear Factors<\/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 your denominator breaks down into separate linear factors (like [latex](x-1)(x+3)(2x-5)[\/latex]), you get the simplest partial fraction setup. Each distinct linear factor gets its own fraction with a constant numerator\u2014no sharing, no complications.<\/p>\r\n<p class=\"whitespace-normal break-words\">If [latex]Q(x) = (a_1x + b_1)(a_2x + b_2)\\cdots(a_nx + b_n)[\/latex] where each factor appears exactly once, then: [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{a_1x + b_1} + \\frac{A_2}{a_2x + b_2} + \\cdots + \\frac{A_n}{a_nx + b_n}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Two Methods to Find the Constants:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Method 1 - Strategic Substitution (Usually Faster):<\/strong> Substitute values of [latex]x[\/latex] that make individual factors zero. This isolates one constant at a time. For [latex]x(x-2)(x+1)[\/latex], try [latex]x = 0[\/latex], [latex]x = 2[\/latex], and [latex]x = -1[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Method 2 - Equating Coefficients (More Systematic):<\/strong> Expand everything, collect like terms, and match coefficients on both sides. This gives you a system of equations to solve.<\/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>Factor the denominator<\/strong> completely<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Set up the partial fraction template<\/strong> (one fraction per distinct factor)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Find the constants<\/strong> using either method<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Integrate each simple fraction<\/strong> separately<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">Always verify your decomposition by adding the fractions back together\u2014you should get your original function.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165042266568\" data-type=\"problem\">\r\n<p id=\"fs-id1165042266570\">Evaluate [latex]\\displaystyle\\int \\frac{x+1}{\\left(x+3\\right)\\left(x - 2\\right)}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558892\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558892\"]\r\n<div id=\"fs-id1165042277051\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165041836267\">[latex]\\frac{x+1}{\\left(x+3\\right)\\left(x - 2\\right)}=\\frac{A}{x+3}+\\frac{B}{x - 2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558893\"]\r\n<div id=\"fs-id1165042305329\" data-type=\"solution\">\r\n<p id=\"fs-id1165042305331\">[latex]\\frac{2}{5}\\text{ln}|x+3|+\\frac{3}{5}\\text{ln}|x - 2|+C[\/latex]<\/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\/STtozLX2gbk?controls=0&amp;start=826&amp;end=953&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions826to953_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.4 Partial Fractions\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Repeated Linear Factors<\/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 a linear factor appears multiple times in your denominator, you can't just use it once in your partial fraction setup. If you have [latex](ax+b)^n[\/latex], you need to include <strong>all powers<\/strong> from [latex]1[\/latex] up to [latex]n[\/latex]. Think of it like building a complete ladder\u2014you can't skip rungs.<\/p>\r\n<p class=\"whitespace-normal break-words\">For a repeated factor [latex](ax+b)^n[\/latex], your decomposition must include: [latex]\\frac{A_1}{ax+b} + \\frac{A_2}{(ax+b)^2} + \\frac{A_3}{(ax+b)^3} + \\cdots + \\frac{A_n}{(ax+b)^n}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Don't forget the lower powers! It is a common mistake to only include the highest power [latex](ax+b)^n[\/latex]\u00a0and miss the essential terms with smaller exponents.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Finding the Constants - Your Two Options:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Strategic Substitution:<\/strong> Choose [latex]x[\/latex]-values that zero out specific factors. Then use any other [latex]x[\/latex]-value (like [latex]x = 0[\/latex]) to find remaining unknowns.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Equating Coefficients:<\/strong> Expand both sides completely, collect like terms by powers of [latex]x[\/latex], and match coefficients. This gives you a system of equations to solve systematically.<\/p>\r\n<p class=\"whitespace-normal break-words\">Remember these key integrals for the final step:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{ax+b} dx = \\frac{1}{a}\\ln|ax+b| + C[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{(ax+b)^n} dx = \\frac{1}{a} \\cdot \\frac{(ax+b)^{1-n}}{1-n} + C[\/latex] for [latex]n \\neq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165042088547\" data-type=\"problem\">\r\n<p id=\"fs-id1165042281590\">Set up the partial fraction decomposition for [latex]\\displaystyle\\int \\frac{x+2}{{\\left(x+3\\right)}^{3}{\\left(x - 4\\right)}^{2}}dx[\/latex]. (Do not solve for the coefficients or complete the integration.)<\/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-id1165042004334\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165041977514\">Use the problem-solving method of the above example for guidance.<\/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-id1165042194352\" data-type=\"solution\">\r\n<p id=\"fs-id1165040665662\">[latex]\\frac{x+2}{{\\left(x+3\\right)}^{3}{\\left(x - 4\\right)}^{2}}=\\frac{A}{x+3}+\\frac{B}{{\\left(x+3\\right)}^{2}}+\\frac{C}{{\\left(x+3\\right)}^{3}}+\\frac{D}{\\left(x - 4\\right)}+\\frac{E}{{\\left(x - 4\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try 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\/STtozLX2gbk?controls=0&amp;start=1372&amp;end=1422&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions1372to1422_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.4 Partial Fractions\" here (opens in new window)<\/a>.\r\n\r\n<\/section>\r\n<h2>Partial Fraction Decomposition<\/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\">Partial fraction decomposition follows a predictable pattern once you know how to read the denominator. Think of it as having a toolkit where each type of factor in the denominator tells you exactly which \"tool\" to use in your decomposition.<\/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>Check the degree requirement<\/strong> - If deg(numerator) \u2265 deg(denominator), use polynomial long division first<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Factor the denominator completely<\/strong> into linear and irreducible quadratic factors<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Set up your partial fraction template<\/strong> based on what factors you found<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Solve for the unknown constants<\/strong> using strategic substitution or equating coefficients<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Integrate each piece<\/strong> using standard techniques<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>The Factor-to-Fraction Translation Guide:<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Distinct Linear Factor:\u00a0<\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax+b)$$ gives \u2026\">[latex](ax+b)[\/latex] gives [latex]\\frac{A}{ax+b}[\/latex]<\/span><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Repeated Linear Factor:\u00a0<\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax+b)^n$$ give\u2026\">[latex](ax+b)^n[\/latex] gives [latex]\\frac{A_1}{ax+b} + \\frac{A_2}{(ax+b)^2} + \\cdots + \\frac{A_n}{(ax+b)^n}[\/latex]<\/span><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Irreducible Quadratic: <\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax^2+bx+c)$$ g\u2026\">[latex](ax^2+bx+c)[\/latex] gives [latex]\\frac{Ax+B}{ax^2+bx+c}[\/latex]<\/span><\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Repeated Irreducible Quadratic: <\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax^2+bx+c)^n$$\u2026\">[latex](ax^2+bx+c)^n[\/latex] gives [latex]\\frac{A_1x+B_1}{ax^2+bx+c} + \\frac{A_2x+B_2}{(ax^2+bx+c)^2} + \\cdots[\/latex]<\/span><\/p>\r\n<p class=\"whitespace-normal break-words\">\u00a0An irreducible quadratic has no real zeros\u2014check this using the discriminant [latex]b^2-4ac &lt; 0[\/latex].<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165041926585\" data-type=\"problem\">\r\n<p id=\"fs-id1165041926587\">Set up the partial fraction decomposition for [latex]\\displaystyle\\int \\frac{{x}^{2}+3x+1}{\\left(x+2\\right){\\left(x - 3\\right)}^{2}{\\left({x}^{2}+4\\right)}^{2}}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558839\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558839\"]\r\n<div id=\"fs-id1165040744484\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1165040744492\">Use the problem-solving strategy.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558849\"]\r\n<div id=\"fs-id1165041889366\" data-type=\"solution\">\r\n<p id=\"fs-id1165041889368\">[latex]\\frac{{x}^{2}+3x+1}{\\left(x+2\\right){\\left(x - 3\\right)}^{2}{\\left({x}^{2}+4\\right)}^{2}}=\\frac{A}{x+2}+\\frac{B}{x - 3}+\\frac{C}{{\\left(x - 3\\right)}^{2}}+\\frac{Dx+E}{{x}^{2}+4}+\\frac{Fx+G}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/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\/STtozLX2gbk?controls=0&amp;start=2068&amp;end=2110&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions2068to2110_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"3.4 Partial Fractions\" here (opens in new window)<\/a>.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Break down and integrate rational functions using partial fractions<\/li>\n<li>Identify and work with simple linear factors in rational functions<\/li>\n<li>Handle repeated linear factors when using partial fractions<\/li>\n<li>Work with quadratic factors in rational functions<\/li>\n<\/ul>\n<\/section>\n<h2>What Is Partial Fraction Decomposition?<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of partial fraction decomposition as taking apart a complicated fraction to reveal the simple pieces hiding inside. Instead of combining fractions like [latex]\\frac{1}{x+1} + \\frac{2}{x-2}[\/latex], you&#8217;re doing the reverse\u2014starting with something messy like [latex]\\frac{3x}{x^2-x-2}[\/latex] and splitting it back into those manageable pieces.<\/p>\n<p>You can only use partial fractions when the degree of the numerator is <strong>less than<\/strong> the degree of the denominator. If it&#8217;s not, you must use polynomial long division first to create a &#8220;proper&#8221; fraction.<\/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>Check the degree requirement<\/strong> &#8211; Is deg(numerator) &lt; deg(denominator)?<\/li>\n<li class=\"whitespace-normal break-words\"><strong>If not, use polynomial long division<\/strong> to get [latex]\\frac{P(x)}{Q(x)} = A(x) + \\frac{R(x)}{Q(x)}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Factor the denominator<\/strong> completely<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Set up partial fractions<\/strong> based on the factors<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Solve for the unknown constants<\/strong><\/li>\n<li class=\"whitespace-normal break-words\"><strong>Integrate each simple fraction<\/strong><\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">You can always verify your decomposition by finding a common denominator and checking that you get back to the original fraction.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165040743552\" data-type=\"problem\">\n<p id=\"fs-id1165042047432\">Evaluate [latex]\\displaystyle\\int \\frac{x - 3}{x+2}dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Hint<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042133890\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165041977487\">Use long division to obtain [latex]\\frac{x - 3}{x+2}=1-\\frac{5}{x+2}[\/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=\"q44558898\">Show Solution<\/button><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041985161\" data-type=\"solution\">\n<p id=\"fs-id1165042050319\">[latex]x - 5\\text{ln}|x+2|+C[\/latex]<\/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\/STtozLX2gbk?controls=0&amp;start=135&amp;end=182&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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions135to182_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.4 Partial Fractions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h2 data-type=\"title\">Nonrepeated Linear Factors<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When your denominator breaks down into separate linear factors (like [latex](x-1)(x+3)(2x-5)[\/latex]), you get the simplest partial fraction setup. Each distinct linear factor gets its own fraction with a constant numerator\u2014no sharing, no complications.<\/p>\n<p class=\"whitespace-normal break-words\">If [latex]Q(x) = (a_1x + b_1)(a_2x + b_2)\\cdots(a_nx + b_n)[\/latex] where each factor appears exactly once, then: [latex]\\frac{P(x)}{Q(x)} = \\frac{A_1}{a_1x + b_1} + \\frac{A_2}{a_2x + b_2} + \\cdots + \\frac{A_n}{a_nx + b_n}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Two Methods to Find the Constants:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Method 1 &#8211; Strategic Substitution (Usually Faster):<\/strong> Substitute values of [latex]x[\/latex] that make individual factors zero. This isolates one constant at a time. For [latex]x(x-2)(x+1)[\/latex], try [latex]x = 0[\/latex], [latex]x = 2[\/latex], and [latex]x = -1[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Method 2 &#8211; Equating Coefficients (More Systematic):<\/strong> Expand everything, collect like terms, and match coefficients on both sides. This gives you a system of equations to solve.<\/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>Factor the denominator<\/strong> completely<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Set up the partial fraction template<\/strong> (one fraction per distinct factor)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Find the constants<\/strong> using either method<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Integrate each simple fraction<\/strong> separately<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">Always verify your decomposition by adding the fractions back together\u2014you should get your original function.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165042266568\" data-type=\"problem\">\n<p id=\"fs-id1165042266570\">Evaluate [latex]\\displaystyle\\int \\frac{x+1}{\\left(x+3\\right)\\left(x - 2\\right)}dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558892\">Hint<\/button><\/p>\n<div id=\"q44558892\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042277051\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165041836267\">[latex]\\frac{x+1}{\\left(x+3\\right)\\left(x - 2\\right)}=\\frac{A}{x+3}+\\frac{B}{x - 2}[\/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=\"q44558893\">Show Solution<\/button><\/p>\n<div id=\"q44558893\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042305329\" data-type=\"solution\">\n<p id=\"fs-id1165042305331\">[latex]\\frac{2}{5}\\text{ln}|x+3|+\\frac{3}{5}\\text{ln}|x - 2|+C[\/latex]<\/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\/STtozLX2gbk?controls=0&amp;start=826&amp;end=953&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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions826to953_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.4 Partial Fractions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h2 data-type=\"title\">Repeated Linear Factors<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When a linear factor appears multiple times in your denominator, you can&#8217;t just use it once in your partial fraction setup. If you have [latex](ax+b)^n[\/latex], you need to include <strong>all powers<\/strong> from [latex]1[\/latex] up to [latex]n[\/latex]. Think of it like building a complete ladder\u2014you can&#8217;t skip rungs.<\/p>\n<p class=\"whitespace-normal break-words\">For a repeated factor [latex](ax+b)^n[\/latex], your decomposition must include: [latex]\\frac{A_1}{ax+b} + \\frac{A_2}{(ax+b)^2} + \\frac{A_3}{(ax+b)^3} + \\cdots + \\frac{A_n}{(ax+b)^n}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Don&#8217;t forget the lower powers! It is a common mistake to only include the highest power [latex](ax+b)^n[\/latex]\u00a0and miss the essential terms with smaller exponents.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Finding the Constants &#8211; Your Two Options:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Strategic Substitution:<\/strong> Choose [latex]x[\/latex]-values that zero out specific factors. Then use any other [latex]x[\/latex]-value (like [latex]x = 0[\/latex]) to find remaining unknowns.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Equating Coefficients:<\/strong> Expand both sides completely, collect like terms by powers of [latex]x[\/latex], and match coefficients. This gives you a system of equations to solve systematically.<\/p>\n<p class=\"whitespace-normal break-words\">Remember these key integrals for the final step:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{ax+b} dx = \\frac{1}{a}\\ln|ax+b| + C[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\int \\frac{1}{(ax+b)^n} dx = \\frac{1}{a} \\cdot \\frac{(ax+b)^{1-n}}{1-n} + C[\/latex] for [latex]n \\neq 1[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165042088547\" data-type=\"problem\">\n<p id=\"fs-id1165042281590\">Set up the partial fraction decomposition for [latex]\\displaystyle\\int \\frac{x+2}{{\\left(x+3\\right)}^{3}{\\left(x - 4\\right)}^{2}}dx[\/latex]. (Do not solve for the coefficients or complete the integration.)<\/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-id1165042004334\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165041977514\">Use the problem-solving method of the above example for guidance.<\/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-id1165042194352\" data-type=\"solution\">\n<p id=\"fs-id1165040665662\">[latex]\\frac{x+2}{{\\left(x+3\\right)}^{3}{\\left(x - 4\\right)}^{2}}=\\frac{A}{x+3}+\\frac{B}{{\\left(x+3\\right)}^{2}}+\\frac{C}{{\\left(x+3\\right)}^{3}}+\\frac{D}{\\left(x - 4\\right)}+\\frac{E}{{\\left(x - 4\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try 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\/STtozLX2gbk?controls=0&amp;start=1372&amp;end=1422&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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions1372to1422_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.4 Partial Fractions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<h2>Partial Fraction Decomposition<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Partial fraction decomposition follows a predictable pattern once you know how to read the denominator. Think of it as having a toolkit where each type of factor in the denominator tells you exactly which &#8220;tool&#8221; to use in your decomposition.<\/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>Check the degree requirement<\/strong> &#8211; If deg(numerator) \u2265 deg(denominator), use polynomial long division first<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Factor the denominator completely<\/strong> into linear and irreducible quadratic factors<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Set up your partial fraction template<\/strong> based on what factors you found<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Solve for the unknown constants<\/strong> using strategic substitution or equating coefficients<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Integrate each piece<\/strong> using standard techniques<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>The Factor-to-Fraction Translation Guide:<\/strong><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Distinct Linear Factor:\u00a0<\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax+b)$$ gives \u2026\">[latex](ax+b)[\/latex] gives [latex]\\frac{A}{ax+b}[\/latex]<\/span><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Repeated Linear Factor:\u00a0<\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax+b)^n$$ give\u2026\">[latex](ax+b)^n[\/latex] gives [latex]\\frac{A_1}{ax+b} + \\frac{A_2}{(ax+b)^2} + \\cdots + \\frac{A_n}{(ax+b)^n}[\/latex]<\/span><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Irreducible Quadratic: <\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax^2+bx+c)$$ g\u2026\">[latex](ax^2+bx+c)[\/latex] gives [latex]\\frac{Ax+B}{ax^2+bx+c}[\/latex]<\/span><\/p>\n<p class=\"whitespace-normal break-words\"><strong>Repeated Irreducible Quadratic: <\/strong><span class=\"katex-error\" title=\"ParseError: KaTeX parse error: Can't use function '$' in math mode at position 1: $\u0332(ax^2+bx+c)^n$$\u2026\">[latex](ax^2+bx+c)^n[\/latex] gives [latex]\\frac{A_1x+B_1}{ax^2+bx+c} + \\frac{A_2x+B_2}{(ax^2+bx+c)^2} + \\cdots[\/latex]<\/span><\/p>\n<p class=\"whitespace-normal break-words\">\u00a0An irreducible quadratic has no real zeros\u2014check this using the discriminant [latex]b^2-4ac < 0[\/latex].<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165041926585\" data-type=\"problem\">\n<p id=\"fs-id1165041926587\">Set up the partial fraction decomposition for [latex]\\displaystyle\\int \\frac{{x}^{2}+3x+1}{\\left(x+2\\right){\\left(x - 3\\right)}^{2}{\\left({x}^{2}+4\\right)}^{2}}dx[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558839\">Hint<\/button><\/p>\n<div id=\"q44558839\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165040744484\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1165040744492\">Use the problem-solving strategy.<\/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=\"q44558849\">Show Solution<\/button><\/p>\n<div id=\"q44558849\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041889366\" data-type=\"solution\">\n<p id=\"fs-id1165041889368\">[latex]\\frac{{x}^{2}+3x+1}{\\left(x+2\\right){\\left(x - 3\\right)}^{2}{\\left({x}^{2}+4\\right)}^{2}}=\\frac{A}{x+2}+\\frac{B}{x - 3}+\\frac{C}{{\\left(x - 3\\right)}^{2}}+\\frac{Dx+E}{{x}^{2}+4}+\\frac{Fx+G}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/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\/STtozLX2gbk?controls=0&amp;start=2068&amp;end=2110&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.You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.4PartialFractions2068to2110_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;3.4 Partial Fractions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":541,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/718"}],"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\/718\/revisions"}],"predecessor-version":[{"id":2226,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/718\/revisions\/2226"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/541"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/718\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=718"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=718"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=718"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=718"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}