Evaluate the definite integrals`int_(pi/2)^pie^x((1-sinx)/(1-cosx))dx`

Explanation

Let
`I= int_(pi//2)^(pi) e^(x) ((1-sin x)/(1-cos x))dx`
`=int_(pi//2)^(pi)e^(x)[(1-2sin((x)/(2))cos ((x)/(2)))/(2sin^(2)((x)/(2)))]dx`
`=int_(pi//2)^(pi) e^(x) ((1)/(2)"cosec"^(2) (x)/(2) -cot .(x)/(2))dx`
`int_(pi//2)^(pi) e^(x) (-cot .(x)/(2) +(1)/(2) " cosex"^(2) .(x)/(2))dx`
`[underset(inte^(x) {f(x) +f(x)}dx =e^(x)f(x))("Here " (d)/(dx) (-cot.(x)/(2))=(1)/(2)"cosec"^(2).(x)/(2))]`
`:. I=[e^(x) (-cot .(x)/(2))]_(pi//2)^(pi)`
`=-e^(x) cot ((pi)/(2))-[-e^(pi//2)cot ((pi)/(4))]`
`=-e^(pi).0+e^(pi//2).1=e^(pi//2)`