Evaluate the following definite integral: `int_(pi//2)^pie^x((1-sin x)/(1-cos x))dx`

Explanation

`I=int_{frac{pi}{2}}^{pi} e^{x}(frac{1-sin x}{1-cos x}) d x `
`
=int_{frac{pi}{2}}^{pi} e^{x}(frac{1-2 sin frac{x}{2} cos frac{x}{2}}{2 sin ^{2} frac{x}{2}}) d x `
`
=int_{frac{pi}{2}}^{pi} e^{x}(frac{{cosec}^{2} frac{pi}{2}}{2}-cot frac{x}{2}) d x`


Take,` f(x)=-cot frac{{x}}{2}`
`


Rightarrow {f}^{prime}({x})=-(-frac{1}{2} {cosec}^{2} frac{{x}}{2})=frac{1}{2} {cosec}^{2 {x}} `
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