{"id":10026,"date":"2022-02-09T22:40:14","date_gmt":"2022-02-09T17:10:14","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10026"},"modified":"2022-02-09T22:40:19","modified_gmt":"2022-02-09T17:10:19","slug":"integration-by-substitution-formula-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-by-substitution-formula-and-examples\/","title":{"rendered":"Integration By Substitution – Formula and Examples"},"content":{"rendered":"

Here you will learn what is integration by substitution method class 12 with examples.<\/p>\n

Let’s begin –<\/p>\n

Integration By Substitution<\/h2>\n

The method of evaluating an integral by reducing it to standard form by a proper substitution is called integration by substitution.<\/p>\n

If \\(\\phi(x)\\) is continuously differentiable function, then to evaluate integrals of the form<\/p>\n

\\(\\int\\) \\(f(\\phi(x))\\) \\(\\phi'(x)\\) dx, we substitute \\(\\phi(x)\\) = t and \\(\\phi'(x)\\) dx = dt<\/p><\/blockquote>\n

This substitution reduces the above integral to \\(\\int\\) f(t) dt.<\/p>\n

After evaluating this integral we substitute back the value of t.<\/p>\n

Also Read<\/strong> : Integration Formulas for Class 12 \u2013 Indefinite Integration<\/a><\/p>\n

Example<\/strong><\/span> : Prove that \\(\\int\\) sin(ax + b) dx = \\(-1\\over a\\) cos(ax + b) + C.<\/p>\n

Solution<\/strong><\/span> : Let ax + b = t. Then, d(ax + b) = dt \\(\\implies\\) a dx = dt \\(\\implies\\) dx = \\(1\\over a\\) dt<\/p>\n

Putting ax + b = t and dx = \\(1\\over a\\) dt, we get<\/p>\n

\\(\\int\\) sin(ax + b) dx = \\(1\\over a\\) \\(\\int\\) sin t dt<\/p>\n

= \\(-1\\over a\\) cos t + C<\/p>\n

= \\(-1\\over a\\) cos(ax + b) + C<\/p>\n

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

Solution<\/strong><\/span> : I = \\(\\int\\) \\((1-sin^2x)cosx\\over {sinx(1 + sinx)}\\) dx = \\(\\int\\) \\(1 – sinx\\over {sinx}\\) cosx dx<\/p>\n

Put sinx = t\u00a0 \\(\\implies\\)\u00a0 cosx dx = dt<\/p>\n

\\(\\implies\\) I = \\(\\int\\) \\(1 – t\\over t\\) dt<\/p>\n

= ln| t | – t + C<\/p>\n

= ln|sinx| – sinx + C<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is integration by substitution method class 12 with examples. Let’s begin – Integration By Substitution The method of evaluating an integral by reducing it to standard form by a proper substitution is called integration by substitution. If \\(\\phi(x)\\) is continuously differentiable function, then to evaluate integrals of the form \\(\\int\\) …<\/p>\n

Integration By Substitution – Formula and Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[30],"tags":[204,438,889],"yoast_head":"\nIntegration By Substitution - Formula and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is integration by substitution method class 12 with examples.\" \/>\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-substitution-formula-and-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration By Substitution - 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