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 ?

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