{"id":5847,"date":"2021-10-03T16:43:05","date_gmt":"2021-10-03T11:13:05","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5847"},"modified":"2021-11-26T00:04:58","modified_gmt":"2021-11-25T18:34:58","slug":"integration-of-sinx","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/integration-of-sinx\/","title":{"rendered":"Integration of Sinx"},"content":{"rendered":"

Here you will learn proof of integration of sinx or sin x and examples based on it.<\/p>\n

Let’s begin –<\/p>\n

Integration of Sinx or Sin x<\/h2>\n
\n

The integration of sinx is -cosx + C<\/b><\/p>\n

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

i.e. \\(\\int\\) (sinx) dx = -cos x + C<\/p>\n<\/blockquote>\n

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

\n

We will prove this formula using differentiation,<\/p>\n

Let \\(d\\over dx\\)(-cos x + C) = -\\(d\\over dx\\) cos x + \\(d\\over dx\\) C<\/p>\n

Using differentiation formula,<\/p>\n

\\(d\\over dx\\) cos x = -sin x and differentation of constant is 0.<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(-cos x + C) = -\\(d\\over dx\\) cos x + \\(d\\over dx\\) C<\/p>\n

\\(\\implies\\) \\(d\\over dx\\)(-cos x + C) = -(-sin x) + 0<\/p>\n

We can also write it as,<\/p>\n

sin x = \\(d\\over dx\\)(-cos x + C)<\/p>\n

Now, integrating on both sides,<\/p>\n

\\(\\int\\) sin x = \\(\\int\\) \\(d\\over dx\\)(-cos x + C)<\/p>\n

We know that integration and differentiation both are reciprocals of each other, so in right hand side expression they cancel each other and we get,<\/p>\n

Hence, \\(\\int\\) sin x = -cos x + C<\/p>\n<\/blockquote>\n

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

Solution<\/span><\/strong> : We have,\u00a0<\/p>\n

I = \\(\\int\\) sin (ax + b) dx<\/p>\n

Let ax + b = t, Then , d(ax + b) = dt \\(\\implies\\) adx = dt<\/p>\n

\\(\\implies\\) dx = \\(1\\over a\\) dt<\/p>\n

Putting ax + b = t and\u00a0 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

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

\n

Related Questions<\/h3>\n

What is the Differentiation of sin x ?<\/a><\/p>\n

What is the Integration of Sin Inverse x ?<\/a><\/p>\n

What is the Differentiation of sin inverse x ?<\/a><\/p>\n\n\n

\n
Next – Integration of Cosx<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn proof of integration of sinx or sin x and examples based on it. Let’s begin – Integration of Sinx or Sin x The integration of sinx is -cosx + C where C is the integration constant. i.e. \\(\\int\\) (sinx) dx = -cos x + C Proof : We will prove this …<\/p>\n

Integration of Sinx<\/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":[431,204,430],"yoast_head":"\nIntegration of Sinx - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn proof of integration of sinx or sin 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-sinx\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integration of Sinx - 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