Using $x=\tan\theta$, find the exact value of \(\displaystyle\int_0^1\dfrac{x^2}{(1+x^2)^2}\,\math

Question

Using $x=\tan\theta$, find the exact value of $\displaystyle\int_0^1\dfrac{x^2}{(1+x^2)^2}\,\mathrm{d}x$.

Accepted Answer

$\dfrac{\pi}{8}-\dfrac{1}{4}$

A Tangent Substitution