Evaluate the following integral: `int_0^1tan^(-1)((2x)/(1-x^2))dx`

Explanation

`int_{0}^{1} tan ^{-1}(frac{2 x}{1-x^{2}}) d x`


Let, `x=tan theta, then d x=sec ^{2} theta d theta`

When, `x to 0 ; theta to 0`

And `x to 1 ; theta to frac{pi}{4}`

Therefore the integral becomes


`int_{0}^{frac{pi}{4}} tan ^{-1}(frac{2 tan theta}{1-tan ^{2} theta}) sec ^{2} theta d theta `

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