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

Explanation

Let `I=int_{0}^{pi}(frac{x}{1+sin ^{2} x}+cos ^{7} x) d x`


Then,



`I=int_{0}^{pi}(frac{pi-x}{1+sin ^{2}(pi-x)}+cos ^{7}(pi-x)) d x `


`=int_{0}^{pi}(frac{pi-x}{1+sin ^{2} x}-cos ^{7} x) d x ldots ldots ldots ldots ldots ldots(2) `


`text { Adding (1) and (2), we get } `


`2 I=int_{0}^{pi}(frac{x}{1+sin ^{2} x}+cos ^{7} x+frac{pi-x}{1+sin ^{2} x}-cos ^{7} x) d x `


`Rightarrow 2 I=pi int_{0}^{pi} frac{1}{1+sin ^{2} x} d x`


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