Evaluate the following integral:`int_(-pi)^pi(2x(1+sinx))/(1+cos^2dx)dx`

Explanation

Given: `int_(-pi)^pi(2x(1+sinx))/(1+cos^2dx)dx`

`I=int_(-pi)^pi(2x(1+sinx))/(1+cos^2dx)dx`

`I=int_(-pi)^pi(2x)/(1+cos^2dx)dx+int_(-pi)^pi(2xsinx)/(1+cos^2dx)dx`

`I=int_(0)^pi(2(pi-x)sinx)/(1+cos^2dx)dx`

`2I=2pi int_(0)^pi(sinx)/(1+cos^2dx)dx`

`I=pi int_(-1)^1 (dt)/(1+t^2)`

`I=pi[tan^(-1)t]_(-1)^1=(pi^2)/2`