Integration of Tan Inverse x

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

Let’s begin –

Integration of Tan Inverse x

The integration of tan inverse x or arctan x is \(xtan^{-1}x\) – \(1\over 2\) \(log |1 + x^2|\) + C

Where C is the integration constant.

i.e. \(\int\) \(tan^{-1}x\) = \(xtan^{-1}x\) – \(1\over 2\) \(log |1 + x^2|\) + C

Proof : 

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

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

Then, x = tan t

\(\implies\) dx = d(tan t) = \(sec^2 t\) dt

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

\(\implies\) I = \(\int\) t \(sec^2 t\) dt

By using integration by parts formula,

I = t tan t – \(\int\) 1. (tan t) dt

I = t tan t + log |cos t| + C

Since tan t = x \(\implies\) cost = \(1\over \sqrt{1 + tan^2 t}\) = \(1\over \sqrt{1 + x^2}\)

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

\(\implies\) I = x \(tan^{-1}x\) + \(log |{1\over \sqrt{ 1+ x^2}}|\) + C

Hence, \(\int\) \(tan^{-1}x\) dx = \(xtan^{-1}x\) – \(1\over 2\) \(log |1 + x^2|\) + C

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

Solution : We have,

I = \(\int\)  \(x tan^{-1} x\) dx

By using integration by parts formula,

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

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

= \(x^2\over 2\) \(tan^{-1} x\) – \(1\over 2\) \(\int\)  1 – \(1\over 1 + x^2\)dx

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

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


Related Questions

What is the Differentiation of tan inverse x ?

What is the Integration of tan x ?

What is the integration of tan inverse root x ?

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