# 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 ?