7.2 definite integral — Maths STD 12 Science — Question

$\int_0^\infty  {\frac{{{x^3}dx}}{{(1 + x){{\left( {1 + {x^2}} \right)}^2}}}} $

put $x=\tan \theta$

$\int_0^{\frac{\pi }{2}} {\frac{{{{\tan }^3}\theta {{\sec }^2}\theta d\theta }}{{(1 + \tan \theta ){{\sec }^2}\theta  \cdot {{\sec }^2}\theta }}} $

$I=\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{3} \theta d \theta}{\cos \theta+\sin \theta}$

apply king

$I=\int_{0}^{\frac{\pi}{2}} \frac{\cos ^{3} \theta d \theta}{\cos \theta+\sin \theta}$

on add $2 \mathrm{I}=\int_{0}^{\pi / 2}\left(\cos ^{2} \theta+\sin ^{2} \theta-\cos \theta \sin \theta\right)$

$=\frac{\pi}{2}+\left[\frac{\cos 2 \theta}{4}\right]_{0}^{\pi / 2}$

$I=\frac{\pi-1}{4}$