# Integration of Sec Inverse x and Cosec Inverse x

Here you will learn proof of integration of sec inverse x  and cosec inverse x.

Let’s begin –

## Integration of Sec Inverse x

The integration of sec inverse x or arcsec x is $$xsec^{-1}x$$ – $$log|x + \sqrt{x^2 – 1}|$$ + C

Where C is the integration constant.

i.e. $$\int$$ $$sec^{-1}x$$ = $$xsec^{-1}x$$ – $$log|x + \sqrt{x^2 – 1}|$$ + C

Proof :

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

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

Then, x = sec t

$$\implies$$ dx = d(sec t) = sec t tan t dt

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

$$\implies$$ I = $$\int$$ t (sec t tan t) dt

By using integration by parts formula,

I = t sec t – $$\int$$ 1. (sec t) dt

I = t sec t – log |sec t + tan t| + C

Since tan t = $$\sqrt{sec^2 t – 1}$$

$$\implies$$ I = t sec t – $$log |sec t + \sqrt{sec^2t – 1}|$$ + C

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

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

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

## Integration of Cosec Inverse x

The integration of cosec inverse x or arccosec x is x$$cosec^{-1}x$$ – $$log|x – \sqrt{x^2 – 1}|$$ + C

Where C is the integration constant.

i.e. $$\int$$ $$cosec^{-1}x$$ = x$$cosec^{-1}x$$ – $$log|x – \sqrt{x^2 – 1}|$$ + C

Proof :

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

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

Then, x = cosec t

$$\implies$$ dx = d(cosec t) = -cosec t cot t dt

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

$$\implies$$ I = $$\int$$ t (-cosec t cot t) dt

By using integration by parts formula,

I = t cosec t – $$\int$$ 1. (cosec t) dt

I = t cosec t – log |cosec – cot t| + C

Since cot t = $$\sqrt{cosec^2 t – 1}$$

$$\implies$$ I = t cosec t – $$log |cosec t – \sqrt{cosec^2t – 1}|$$ + C

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

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

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

### Related Questions

What is the Differentiation of cosec inverse x ?

What is the Differentiation of sec inverse x ?

What is the integration of sec inverse root x ?