Question

$\int \frac{dx}{(1+x^2{)}^2}$

Answer 1

$Let,I=\int \frac{dx}{{(1+x^2)}^2}wewillusethesubstitutionx=\tan \theta \Rightarrow dx={\sec }^2\theta d\theta =\int \frac{{\sec }^2\theta d\theta }{{\sec }^4\theta }[\because 1+{\tan }^2\theta ={\sec }^2\theta ]=\int \frac{d\theta }{{\sec }^2\theta }=\int {\cos }^2\theta d\theta =\frac{1}{2}\int \cos 2\theta d\theta +\int \frac{1}{2}d\theta [\because \cos 2\theta =2{\cos }^2\theta -1\Rightarrow {\cos }^2\theta =\frac{1}{2}\cos 2\theta +\frac{1}{2}]=\frac{1}{4}\sin 2\theta +\frac{1}{2}\theta +C$