# Integration of Sin Inverse x

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

Let’s begin –

## Integration of Sin Inverse x

The integration of sin inverse x or arcsin x is $$xsin^{-1}x$$ + $$\sqrt{1 – x^2}$$ + C

Where C is the integration constant.

i.e. $$\int$$ $$sin^{-1}x$$ = $$xsin^{-1}x$$ + $$\sqrt{1 – x^2}$$ + C

Proof :

We have, I = $$\int$$ $$sin^{-1}x$$ dx

Let $$sin^{-1}x$$ = t,

Then, x = sint

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

$$\therefore$$ I = $$\int$$ $$sin^{-1}x$$ dx

$$\implies$$ I = $$\int$$ t cost dt

By using integration by parts formula,

I = t sin t – $$\int$$ 1. (sin t) dt

I = t sin t – $$\int$$ sint dt

= t sin t + cos t + C

= t sin t + $$\sqrt{1 – sin^2 t}$$ + C

Now, Put t = $$sin^{-1}x$$ here

$$\implies$$ I = x $$sin^{-1}x$$ + $$\sqrt{1 – x^2}$$ + C

Hence, $$\int$$ $$sin^{-1}x$$ dx = x $$sin^{-1}x$$ + $$\sqrt{1 – x^2}$$ + C

Example : Evaluate $$\int$$ $$x sin^{-1} x$$ dx

Solution : We have,

I = $$\int$$  $$x sin^{-1} x$$ dx

By using integration by parts formula,

I = $$sin^{-1} x$$ $$x^2\over 2$$ – $$\int$$ $$1\over \sqrt{1 – x^2}$$ $$\times$$ $$x^2\over 2$$ dx

I =  $$x^2\over 2$$ $$sin^{-1} x$$ + $$1\over 2$$ $$\int$$ $$-x^2\over \sqrt{1 – x^2}$$ dx

= $$x^2\over 2$$ $$sin^{-1} x$$ + $$1\over 2$$ $$\int$$ $$1 – x^2 – 1\over \sqrt{1 – x^2}$$  dx

= $$x^2\over 2$$ $$sin^{-1} x$$ + $$1\over 2$$ { $$\int$$ $$1 – x^2\over \sqrt{1 – x^2}$$ – $$\int$$ $$1\over \sqrt{1 -x^2}$$ } dx

$$\implies$$ I = $$x^2\over 2$$ $$sin^{-1} x$$ + $$1\over 2$$ { $$\int$$ $$\sqrt{1 – x^2}$$ – $$\int$$ $$1\over \sqrt{1 -x^2}$$ } dx

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

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

$$\implies$$ I = $$x^2\over 2$$ $$sin^{-1} x$$ +  $$1\over 4$$ $$x\sqrt{1 – x^2}$$ – $$1\over 4$$ $$sin^{-1} x$$ + C

### Related Questions

What is the Differentiation of sin inverse x ?

What is the Integration of sin x ?

What is the integration of sin inverse root x ?