Question

${\int }_0^1\frac{xta{n}^{-1}x}{{(1+x^2)}^{3/2}}dx$

Answer 1

$I={\int }_0^1\frac{x{\tan }^{-1}x}{{\left(1+x^2\right)}^{\frac{3}{2}}}dx$

$put{\tan }^{-1}x=t$

$\frac{1}{\left(1+x^2\right)}dx=dt$

$x\to 0$ $\Rightarrow$ $t\to 0$ and $x\to 1$ $\Rightarrow$ $t\to \frac{\pi }{4}$

$I={\int }_0^{\frac{\pi }{4}}\frac{\tan t.tdt}{\sec t}$

$={\int }_0^{\frac{\pi }{4}}t\sin tdt$

$={[-x\cos x+\sin x]}_0^{\pi /4}$

$=\frac{1}{\sqrt{2}}-\frac{\pi }{4}\times \frac{1}{\sqrt{2}}$

$=\frac{1}{\sqrt{2}}\left(1-\frac{\pi }{4}\right)$