Integration of Cos Inverse x

Here you will learn proof of integration of cos inverse x or arccos x and examples based on it.

Let’s begin –

Integration of Cos Inverse x

The integration of cos inverse x or arccos x is \(xcos^{-1}x\) – \(\sqrt{1 – x^2}\) + C

Where C is the integration constant.

i.e. \(\int\) \(cos^{-1}x\) = \(xcos^{-1}x\) – \(\sqrt{1 – x^2}\) + C

Proof : 

We have, I = \(\int\) \(cos^{-1}x\) dx

Let \(cos^{-1}x\) = t,

Then, x = cos t

\(\implies\) dx = d(cos t) = -sin t dt

\(\therefore\) I = \(\int\) \(cos^{-1}x\) dx

\(\implies\) I = \(\int\) -t sint dt

By using integration by parts formula,

I = t cos t + \(\int\) 1. (-cos t) dt

I = t cos t – \(\int\) cost dt

= t cos t – sin t + C

= t cos t – \(\sqrt{1 – cos^2 t}\) + C

Now, Put t = \(cos^{-1}x\) here

\(\implies\) I = x \(cos^{-1}x\) – \(\sqrt{1 – x^2}\) + C

Hence, \(\int\) \(cos^{-1}x\) dx = x \(cos^{-1}x\) – \(\sqrt{1 – x^2}\) + C

Example : Evaluate \(\int\) \(x cos^{-1} x\) dx

Solution : We have,

I = \(\int\)  \(x cos^{-1} x\) dx

By using integration by parts formula,

I = \(cos^{-1} x\) \(x^2\over 2\) – \(\int\) \(-1\over \sqrt{1 – x^2}\) \(\times\) \(x^2\over 2\) dx

I =  \(x^2\over 2\) \(cos^{-1} x\) – \(1\over 2\) \(\int\) \(-x^2\over \sqrt{1 – x^2}\) dx

= \(x^2\over 2\) \(cos^{-1} x\) – \(1\over 2\) \(\int\) \(1 – x^2 – 1\over \sqrt{1 – x^2}\)  dx

= \(x^2\over 2\) \(cos^{-1} x\) – \(1\over 2\) { \(\int\) \(1 – x^2\over \sqrt{1 – x^2}\) – \(\int\) \(1\over \sqrt{1 -x^2}\) } dx

\(\implies\) I = \(x^2\over 2\) \(cos^{-1} x\) – \(1\over 2\) { \(\int\) \(\sqrt{1 – x^2}\) – \(\int\) \(1\over \sqrt{1 -x^2}\) } dx

By using integration formula of \(\sqrt{a^2 – x^2}\),

\(\implies\) I = \(x^2\over 2\) \(cos^{-1} x\) – \(1\over 2\) [{ \(1\over 2\) \(x\sqrt{1 – x^2}\) – \(1\over 2\) \(sin^{-1} x\) } – \(sin ^{-1} x\) ] + C

\(\implies\) I = \(x^2\over 2\) \(cos^{-1} x\) –  \(1\over 4\) \(x\sqrt{1 – x^2}\) + \(3\over 4\) \(sin^{-1} x\) + C


Related Questions

What is the Differentiation of cos inverse x ?

What is the Integration of cos x ?

What is the integration of cos inverse root x ?

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