{"id":186,"date":"2019-08-27T18:14:09","date_gmt":"2019-08-27T18:14:09","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=186"},"modified":"2019-12-02T21:35:15","modified_gmt":"2019-12-02T21:35:15","slug":"partial-fractions","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/partial-fractions\/","title":{"rendered":"Partial Fractions"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Partial Fractions &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p> The integrand is an <strong>improper rational function<\/strong>.  By &#8220;long division&#8221; of polynomials, we can rewrite the integrand as the sum of a polynomial and a <strong>proper rational function<\/strong> &#8220;remainder&#8221;: <\/p>\n\n\n<style>.kt-accordion-id_3b7c96-44 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_3b7c96-44 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_3b7c96-44 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_3b7c96-44 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_3b7c96-44 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) 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.kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_3b7c96-44 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_3b7c96-44 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3b7c96-44:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_3b7c96-44 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_3b7c96-44 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_3b7c96-44 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_a4cd22-58\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Improper Rational Function<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<p>\n\n An <strong>improper rational function<\/strong> is a ratio $\\displaystyle\\frac{P(x)}{Q(x)}$ of polynomials in which the degree of $P(x)$ is greater than or equal to the degree of $Q(x)$.\n\n<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane_361965-a8\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Proper Rational Function<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<p>\n\nA <strong>proper rational function<\/strong> is a ratio $\\displaystyle\\frac{P(x)}{Q(x)}$ of polynomials in which the degree of $P(x)$ is less than the degree of $Q(x)$.  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>\nNotice that\n\\[\\frac{-1}{x^2-x-6}=\\frac{-1}{(x+2)(x-3)}\\]\nwhich suggests that we try to write $\\displaystyle \\frac{-1}{x^2-x-6}$\nas the sum of two rational functions of the form\n$\\displaystyle\\frac{A}{x+2}$ and $\\displaystyle\\frac{B}{x-3}$:\n\\[\\frac{-1}{x^2-x-6}=\\frac{A}{x+2}+\\frac{B}{x-3}.\\]\nThis is called the <b>Partial Fraction Decomposition<\/b> for\n$\\displaystyle \\frac{-1}{x^2-x-6}$. \n\n<\/p>\n\n\n\n<p>\nOur goal now is to determine $A$ and $B$.  Multiplying both sides of\nthe equation by $(x+2)(x-3)$ to clear the fractions,\n\\[-1=A(x-3)+B(x+2).\\]\nThere are two methods for solving for $A$ and $B$:\n\n<\/p>\n\n\n\n<p>\n\t\tCollect like terms on the right: <br>\n\t\t$-1=(A+B)x+(-3A+2B).$ <br>\n\t\tNow equate coefficients of <br>\n\t\tcorresponding powers of $x$: <br>\n\t\t$A+B=0,\\quad -3A+2B=-1.$ <br>\n\t\tSolving this system, <br>\n\t\t$A=1\/5$, $B=-1\/5$.\n<\/p>\n\n\n\n<p>\n\t\tThe equation holds for <i>all<\/i> $x$. <br>\n\t\tLet $x=-2$: <br>\n\t\t$-1=A(-2-3)+B(-2+2)$ <br>\n\t\t$-1=-5A\\qquad\\longrightarrow A=1\/5$. <br>\n\t\tNow let $x=3$: <br>\n\t\t$-1=A(3-3)+B(3+2)$ <br>\n\t\t$-1=5B\\qquad\\longrightarrow B=-1\/5$.\n<br><\/p>\n\n\n\n<p>\nSo \n\\[\\frac{-1}{x^2-x-6}=\\frac{\\frac{1}{5}}{x+2}-\\frac{\\frac{1}{5}}{x-3}.\\]\nReturning to the original integral,\n\\begin{eqnarray*}\n\t\\int \\frac{3x^3-2x^2-19x-7}{x^2-x-6}\\, dx&amp;=&amp;\\int\n\t\\left(3x+1+\\frac{\\frac{1}{5}}{x+2}-\\frac{\\frac{1}{5}}{x-3}\\right)\\, dx\\\\\n\t&amp;=&amp;\\frac{3}{2}x^2+x+\\frac{1}{5}\\ln \\left|\\frac{x+2}{x-3}\\right|+C.\n\\end{eqnarray*}\n\n<\/p>\n\n\n\n<p> In the next example, we have repeated factors in the denominator, as well as an <strong>irreducible quadratic factor<\/strong>. <\/p>\n\n\n<style>.kt-accordion-id_255ecf-e9 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_255ecf-e9 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_255ecf-e9 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > 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.kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_255ecf-e9 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_255ecf-e9 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_255ecf-e9:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_255ecf-e9 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_255ecf-e9 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_255ecf-e9 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_d5f424-2a\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Irreducible Quadratic Factor<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n\nAn <strong>irreducible<\/strong> quadratic polynomial $x^2+\\beta x+\\gamma$ cannot be factored into linear factors with real coefficients.  That is, $x^2+\\beta x+\\gamma$ has complex roots.  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\nWe will evaluate \n\\[\\int \\frac{x-1}{x^2(x^2+x+1)}\\, dx.\\]\nThe integrand is a proper rational function, which we would like to\ndecompose into proper rational functions of the form \n\\[\\frac{A}{x},\\quad \\frac{B}{x^2},\\quad \\mathrm{and}\\quad \n\\frac{Cx+D}{x^2+x+1}.\\]\n[Notice that we have two\nfactors of $x$ in the denominator of the integrand, leading to terms\nof the form $\\displaystyle\\frac{A}{x}$ and\n$\\displaystyle\\frac{B}{x^2}$ in the decomposition.\nThe factor $x^2+x+1$ is irreducible and quadratic, so any proper\nrational function with $x^2+x+1$ as denominator has the form\n$\\displaystyle\\frac{Cx+D}{x^2+x+1}$ where $C$ or $D$ may be $0$.]\n\n<\/p>\n\n\n\n<p>\nSet\n\\[\\frac{x-1}{x^2(x^2+x+1)}=\\frac{A}{x}+\\frac{B}{x^2}+\\frac{Cx+D}{x^2+x+1}. \\]\nMultiplying through by $x^2(x^2+x+1)$,\n\\[x-1=Ax(x^2+x+1)+B(x^2+x+1)+(Cx+D)x^2.\\]\nSince $x^2+x+1$ has no real roots, it is easiest to solve for $A$ and\n$B$ using Method 1:\n\n<\/p>\n\n\n\n<p>\nCollecting like terms on the right,\n\\[x-1=(A+C)x^3+(A+B+D)x^2+(A+B)x+B.\\]\nEquating corresponding powers of $x$,\n\\[\n\\left.\\begin{array}{rcr}\n\tA+C&amp;=&amp;0\\\\\n\tA+B+D&amp;=&amp;0\\\\\n\tA+B&amp;=&amp;1\\\\\n\tB&amp;=&amp;-1\n\\end{array}\\right\\}\\quad\\longrightarrow\\quad\n\\begin{array}{l}\n\tA=2\\\\\n\tB=-1\\\\\n\tC=-2\\\\\n\tD=-1\n\\end{array}\\quad\\longrightarrow\\quad\n\\frac{2}{x}-\\frac{1}{x^2}-\\frac{2x+1}{x^2+x+1}.\n\\]\nSo\n\\begin{eqnarray*}\n\t\\frac{x-1}{x^2(x^2+x+1)}\\,\n\tdx&amp;=&amp;\\int\\left(\\frac{2}{x}-\\frac{1}{x^2}-\\frac{2x+1}{x^2+x+1}\\right)\\,\n\tdx\\\\\n\t&amp;=&amp;2\\ln |x|+\\frac{1}{x}-\\ln |x^2+x+1|+C\\\\\n\t&amp;=&amp;\\frac{1}{x}+\\ln\\left|\\frac{x^2}{x^2+x+1}\\right|+C.\n\\end{eqnarray*} \n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h3>Key Concepts<\/h3>\n<p>\n<\/p><\/center>\n\n\n\n<h4 class=\"wp-block-heading\">Partial Fraction Decomposition of a Rational Function<\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li> If the rational function is improper, use &#8220;long division&#8221; of\n\t\tpolynomials to write it as the sum of a polynomial and a proper\n\t\trational function &#8220;remainder.&#8221;\n\t\n\t<br><br>\n\t<\/li><li> Decompose the proper rational function as a sum of rational\n\t\tfunctions of the form\n\t\t\\[\\frac{A}{(x-\\alpha)^k} \\quad\\mathrm{and}\\quad \\frac{Bx+C}{(x^2+\\beta\n\t\tx+\\gamma)^k}\\quad (x^2+\\beta x+\\gamma \\mathrm{~irreducible})\\]\n\t\twhere:\n\t\t<ul>\n\t\t\t<li> Each factor $(x-\\alpha)^m$ in the denominator of the proper\n\t\t\t\trational function suggests terms\n\t\t\t\t\\[\\frac{A_1}{(x-\\alpha)}+\\frac{A_2}{(x-\\alpha)^2}+\\ldots\n\t\t\t\t+\\frac{A_m}{(x-\\alpha)^m}.\\]\n\t\t\t\n\t\t\t<\/li><li> Each factor $(x^2+\\beta x+\\gamma)^n$ suggests terms\n\t\t\t\t\\[\\frac{B_1x+C_1}{(x^2+\\beta x+\\gamma)}+\\frac{B_2x+C_2}{(x^2+\\beta\n\t\t\t\tx+\\gamma)^2}+\\ldots +\\frac{B_nx+C_n}{(x^2+\\beta x+\\gamma)^n}.\\]\n\t\t<\/li><\/ul>\n\n\t<\/li><li> Determine the (unique) values of all the constants involved.\n\n\t\t<ul>\n\t\t\t<br>\n\t\t\t<li> Use either Method 1 or Method 2, or a combination of both.\n\t\t<\/li><\/ul>\n<\/li><\/ul>\n\n\n\n<p>\nThe partial fraction decomposition is often used to rewrite a\ncomplicated rational function integrand as a sum of terms, each of\nwhich is straightforward to integrate.\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ0210\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Partial Fractions &#8211; HMC Calculus Tutorial The integrand is an improper rational function. By &#8220;long division&#8221; of polynomials, we can rewrite the integrand as the sum of a polynomial and a proper rational function &#8220;remainder&#8221;: Notice that \\[\\frac{-1}{x^2-x-6}=\\frac{-1}{(x+2)(x-3)}\\] which suggests that we try to write $\\displaystyle \\frac{-1}{x^2-x-6}$ as the sum of two rational functions of&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-186","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/186","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=186"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/186\/revisions"}],"predecessor-version":[{"id":1109,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/186\/revisions\/1109"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=186"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}