{"id":4068,"date":"2021-08-15T19:39:13","date_gmt":"2021-08-15T19:39:13","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4068"},"modified":"2021-11-26T01:25:51","modified_gmt":"2021-11-25T19:55:51","slug":"integration-by-partial-fraction","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-by-partial-fraction\/","title":{"rendered":"Integration by Partial Fraction Formula"},"content":{"rendered":"

Here you will learn integration by partial fraction formula and integration of irrational functions.<\/p>\n

Let’s begin –<\/p>\n

Integration by Partial Fraction Formula<\/h2>\n

(i) Integration of Rational Functions<\/strong><\/p>\n\n\n\n \n \n \n \n
S.No<\/th>\n form of rational function<\/th>\n form of partial fraction<\/th>\n <\/tr>\n
1<\/td>\n \\(px^2+qx+r\\over {(x-a)(x-b)(x-c)}\\)<\/td>\n \\(A\\over {x-a}\\) + \\(B\\over {x-b}\\) + \\(C\\over {x-c}\\)<\/td>\n <\/tr>\n
2<\/td>\n \\(px^2+qx+r\\over {{(x-a)}^2(x-b)}\\)<\/td>\n \\(A\\over {x-a}\\) + \\(B\\over {(x-a)}^2\\) + \\(C\\over {x-b}\\)<\/td>\n <\/tr>\n
3<\/td>\n \\(px^2+qx+r\\over {(x-a)(x^2+bx+c)}\\)<\/td>\n \\(A\\over {x-a}\\) + \\(Bx+C\\over {x^2+bx+c}\\)<\/td>\n <\/tr>\n <\/tbody><\/table>
\n\n\n\n

Example : <\/span> Evaluate \\(\\int\\) \\(x\\over {(x-2)(x-5)}\\) dx<\/p>\n

Solution : <\/span>We have, \\(\\int\\) \\(x\\over {(x-2)(x-5)}\\) dx

\nLet \\(x\\over {(x-2)(x-5)}\\) = \\(A\\over {x-2}\\) + \\(B\\over {x-5}\\)

\n or   x = A(x+5) + B(x-2)

\n by comparing the coefficients, we get

\n A = 2\/7 and B = 5\/7 so that

\n \\(\\int\\) \\(x\\over {(x-2)(x-5)}\\) dx = \\(2\\over 7\\) \\(\\int\\)\\(dx\\over x-2\\) + \\(5\\over 7\\) \\(\\int\\)\\(dx\\over x+5\\)

\n = \\(2\\over 7\\) ln|x-2| + \\(5\\over 7\\) ln|x+5| + C

\n <\/p>\n\n\n\n

Example : <\/span> Evaluate \\(\\int\\) \\(2x\\over {(x^2+1)(x^2+2)}\\) dx<\/p>\n

Solution : <\/span>Let I = \\(\\int\\) \\(2x\\over {(x^2+1)(x^2+2)}\\) dx

\nPutting \\(x^2\\) = t and 2xdx = dt, we get

\nI = \\(\\int\\) \\(dt\\over {(t+1)(t+2)}\\)

\nLet \\(1\\over {(t+1)(t+2)}\\) = \\(A\\over t+1\\) + \\(B\\over t+2\\) …….(i)

\n \\(\\implies\\) 1 = A(t+2) + B(t+1) ……..(ii)

\n Putting t = -2 in (ii), we obtain B = -1

\nPutting t = -1 in (ii), we obtain A = 1

\n Putting value of A and B in (i), we get

\n \\(1\\over {(t+1)(t+2)}\\) = \\(1\\over t+1\\) – \\(1\\over t+2\\)

\n I = \\(\\int\\) \\(1\\over {(t+1)(t+2)}\\)

\n\\(\\implies\\) I = \\(\\int\\) \\(1\\over t+1\\)dt – \\(\\int\\) \\(1\\over t+2\\)dt

\n\\(\\implies\\) I = log|t+1| – log|t+2| + C

\n\\(log|x^2+1|\\) – \\(log|x^2+2|\\) + C

\n <\/p>\n\n\n

(ii)  Integration of Irrational Functions<\/strong><\/p>\n

(a) \\(\\int\\) \\(dx\\over {(ax + b)\\sqrt{px+q}}\\) & \\(\\int\\) \\(dx\\over {(ax^2 + bx + c)\\sqrt{px+q}}\\); put px+q = \\(t^2\\)<\/p>\n

(b)  \\(\\int\\) \\(dx\\over {(ax + b)\\sqrt{px^2+qx+r}}\\); put ax+b = \\(1\\over t\\); \\(\\int\\) \\(dx\\over {(ax^2 + b)\\sqrt{px^2+q}}\\); put x = \\(1\\over t\\) <\/p>\n\n\n

\n
Previous – Integration of Trigonometric Function<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn integration by partial fraction formula and integration of irrational functions. Let’s begin – Integration by Partial Fraction Formula (i) Integration of Rational Functions S.No form of rational function form of partial fraction 1 \\(px^2+qx+r\\over {(x-a)(x-b)(x-c)}\\) \\(A\\over {x-a}\\) + \\(B\\over {x-b}\\) + \\(C\\over {x-c}\\) 2 \\(px^2+qx+r\\over {{(x-a)}^2(x-b)}\\) \\(A\\over {x-a}\\) + \\(B\\over {(x-a)}^2\\) …<\/p>\n

Integration by Partial Fraction Formula<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[30],"tags":[204,432,433],"yoast_head":"\nIntegration by Partial Fraction Formula<\/title>\n<meta name=\"description\" content=\"In this post you will learn method of integration by partial fraction formula with examples and integration of irrational functions.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration by Partial Fraction Formula\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn method of integration by partial fraction formula with examples and integration of irrational functions.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-15T19:39:13+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-25T19:55:51+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Integration by Partial Fraction Formula\",\"datePublished\":\"2021-08-15T19:39:13+00:00\",\"dateModified\":\"2021-11-25T19:55:51+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/\"},\"wordCount\":336,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"integration\",\"integration by partial fractions\",\"integration by partial fractions class 12\"],\"articleSection\":[\"Indefinite Integration\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/\",\"url\":\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/\",\"name\":\"Integration by Partial Fraction Formula\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-08-15T19:39:13+00:00\",\"dateModified\":\"2021-11-25T19:55:51+00:00\",\"description\":\"In this post you will learn method of integration by partial fraction formula with examples and integration of irrational functions.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/integration-by-partial-fraction\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Integration by Partial Fraction Formula\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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