{"id":5835,"date":"2021-10-03T19:08:07","date_gmt":"2021-10-03T13:38:07","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5835"},"modified":"2021-11-25T23:29:25","modified_gmt":"2021-11-25T17:59:25","slug":"integration-of-log-x","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-log-x\/","title":{"rendered":"What is the Integration of Log x dx ?"},"content":{"rendered":"

Here you will learn what is the integration of log x dx with respect to x and examples based on it.<\/p>\n

Let’s begin –<\/p>\n

Integration of Log x<\/h2>\n
\n

The integration of log x with respect to x is x(log x) – x + C<\/strong>.<\/p>\n

where C is the integration Constant.<\/p>\n

i.e. \\(\\int\\) log x dx = x(log x) – x + C<\/p>\n<\/blockquote>\n

Proof :<\/strong><\/p>\n

\n

We will use integration by parts formula<\/a> to prove this,<\/p>\n

Let I = \\(\\int\\) log x .1 dx<\/p>\n

where log x is the first function and 1 is the second function according to ilate rule.<\/p>\n

I = log x . {\\(\\int\\) 1 dx} – \\(\\int\\) { \\(d\\over dx\\) (log x) . \\(\\int\\) 1 dx } dx<\/p>\n

I = (log x) x – \\(\\int\\) \\(1\\over x\\).x dx\u00a0<\/p>\n

= x (log x) – \\(\\int\\) 1 dx<\/p>\n

= x (log x) – x + C<\/p>\n

Hence, \\(\\int\\) log x = x (log x) – x + C<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Evaluate : \\((log x)^2\\) dx<\/p>\n

Solution<\/span><\/strong> : We have,<\/p>\n

I = \\((log x)^2\\) . 1 dx, Then ,<\/p>\n

where \\((log x)^2\\) is the first function and 1 is the second function according to ilate rule,<\/p>\n

I = \\((log x)^2\\) { \\(\\int\\) 1 dx} – \\(\\int\\) {\\(d\\over dx\\) \\((log x)^2\\) . \\(\\int\\) 1 dx } dx<\/p>\n

= \\((log x)^2\\) x – \\(\\int\\) 2 log x . \\(1\\over x\\) . x dx<\/p>\n

= x \\((log x)^2\\) – 2 \\(\\int\\) log x .1 dx<\/p>\n

\\(\\implies\\) I = x \\((log x)^2\\) – 2[ log x { \\(\\int\\) 1 dx } – \\(\\int\\) { \\(d\\over dx\\) (log x) \\(\\int\\) 1 dx } dx ]<\/p>\n

\\(\\implies\\) I = x \\((log x)^2\\) – 2 { (log x) x – \\(\\int\\) \\(1\\over x\\) x dx }<\/p>\n

Hence, I = x( \\((log x)^2\\) – 2 (x log x – x) + C<\/p>\n

\n

Related Questions<\/h3>\n

What is the integration of log cos x dx ?<\/a><\/p>\n

What is the integration of log 1\/x ?<\/a><\/p>\n

What is the integration of 1\/x log x ?<\/a><\/p>\n

What is the integration of x log x dx ?<\/a><\/p>\n\n\n

\n
Next – Integration of Sin Inverse x<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Integration of Cosecx<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the integration of log x dx with respect to x and examples based on it. Let’s begin – Integration of Log x The integration of log x with respect to x is x(log x) – x + C. where C is the integration Constant. i.e. \\(\\int\\) log x dx …<\/p>\n

What is the Integration of Log x dx ?<\/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":[419,204,418],"yoast_head":"\nWhat is the Integration of Log x dx ? - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the integration of log x dx with respect to x and examples based on it.\" \/>\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-of-log-x\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the Integration of Log x dx ? 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