`int_((pi)/(2))^(pi)e^(x)((1-sinx)/(1-cosx))dx`

Explanation

माना `I=int_(pi//2)^(2)e^(x)((1-sinx)/(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)"cosec"^(2)(x)/(2))dx`
`[{:("यहाँ "(d)/(dx)(-cot.(x)/(2))=(1)/(2)"cosec"^(2)(x)/(2)),(inte^(x){f(x)+f'(x)}dx=e^(x)f(x)):}]`
`therefore" "I=[e^(x)(-cot.(x)/(2))]_(pi//2)^(pi)`
`=-e^(pi)cot((pi)/(2))-[-e^(pi//2)cot((pi)/(4))]`
`=e^(pi).0+e^(pi//2).1=e^(pi//2)`