Evaluate the following integral: `int_0^(pi//2)x/(sinx+cos x)dx`

Explanation

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

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

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

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

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I=frac{pi}{4} int_{0}^{pi / 2} frac{1}{sin x+cos x} d x`

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I=frac{pi}{4} times frac{1}{sqrt{2}} int_{0}^{x / 2} frac{1}{sin x cos frac{pi}{4}+sin frac{pi}{4} cos x}`

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I=frac{pi}{4 sqrt{2}} int_{0}^{pi / 2} {cosec}(x+frac{pi}{4}) d x`

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I=frac{pi}{4 sqrt{2}}[ln [{cosec}(x+frac{pi}{4})-cot (x+frac{pi}{4})]]_{0}^{pi / 2}`

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