Evaluate the following integrals: (1-35) `int_0^pi(x/(1+sin^2x)+cos^7x)dx`

Explanation

Given,
I=`int_0^pi(x/(1+sin^2x)+cos^7x)dx....(1)`

I=`int_0^pi((pi-x)/(1+sin^2(pi-x))+cos^7(pi-x))dx....(2)`

adding (1) and (2) we get

`2I=int_0^pi(x/(1+sin^2x)+cos^7x+(pi-x)/(1+sin^2x)-cos^7x)dx`

`2I=pi int_0^(pi)(1/(1+sin^2x))dx`

Dividing numerator and denominator by `cos^2x`

`2I=pi int_0^pi(sec^2x/(sec^2x+tan^2x))dx
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