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

Explanation

Let `I=int_{0}^{pi / 2} frac{sin ^{n} x}{sin ^{n} x+cos ^{n} x} d x quad ldots(1)` So, `I=int_{0}^{pi / 2} frac{sin ^{n}(frac{pi}{2}-x)}{sin ^{n}(frac{pi}{2}-x)+cos ^{n}(frac{pi}{2}-x)} d x quad[because int_{0}^{a} f(x) d x=int_{0}^{a} f(a-x) d x]`

`Rightarrow I=int_{0}^{pi / 2} frac{cos ^{n} x}{sin ^{n} x+cos ^{n} x} d x quad ldots { (2) }`


Adding (1) and (2), we get


`2I=int_{0}^{pi / 2} frac{sin ^{n} x}{sin ^{n} x+cos ^{n} x} d x+int_{0}^{pi / 2} frac{cos ^{n} x}{sin ^{n} x+cos ^{n} x} d x `

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

`Rightarrow 2I=int_{0}^{pi / 2} d x `

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