Question

Evaluate the integral ${\int }_0^{\frac{\pi }{2}}\frac{\sin x}{1+{\cos }^{2}x}dx$ using substitution.

Answer 1

${\int }_0^{\frac{\pi }{2}}\frac{\sin x}{1+{\cos }^{2}x}dx$
Let $\cos x=t\Rightarrow -\sin xdx=dt$
When $x=0,t=1$ and when $x=\frac{\pi }{2},t=0$
$\Rightarrow {\int }_0^{\frac{\pi }{2}}\frac{\sin x}{1+{\cos }^{2}x}dx={\int }_1^0\frac{dt}{1+t^2}$
$=-[{\tan }^{-1}t{]}_1^0$
$=-[{\tan }^{-1}0-{\tan }^{-1}1]$
$=-[-\frac{\pi }{4}]=\frac{\pi }{4}$