Evaluate the following :
`int_(pi//2)^(pi)e^(x)((1-sinx)/(1-cosx))dx.`

Explanation

Let `I=int_(pi//2)^(pi)e^(x)((1-sinx)/(1-cosx))dx`
`=int_(pi//2)^(pi)e^(x)((1)/(1-cosx)-(sinx)/(1-cosx))dx`
`=int_(pi//2)^(pi)e^(x)[(1)/(2sin^(2)x(pi//2))-(2 sin(pi//2).cos(pi//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)"cosec"^(2)((x)/(2))]dx`
Put `f(x)=cot((x)/(2))`
`therefore f'(x)=-"cosec"^(2)((x)/(2))xx(1)/(2)=-(1)/(2)"cosec"^(2)((x)/(2))`
`therefore I=-int_(pi//2)^(pi)e^(x)[f(x)+f'(x)]dx`
`=-[e^(x).f(x)]_(pi//2)^(pi)=-[e^(x).cot((x)/(2))]_(pi//2)^(pi)`
`=-[e^(pi).cot.(pi)/(2)-e^(pi//2).cot.(pi)/(4)]`
`=-[e^(pi)xx0 -e^(pi//2)xx1]=e^(pi//2).`