Question

Evaluate the definite integral:

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

Answer 1

Let I = `int e^x ((1 - sin x)/(1 + cos x))`dx

`= int e^x [(1 - 2 sin  x/2 cos  x/2)/(2 sin^2  x/2)]`dx

`= int e^x (1/2 cosec^2 * x/2 - cot  x/2)`dx

`= - int cot  x/2 e^x dx + 1/2 int e^x cosec^2  x/2  dx`

`= - [cot  x/2 * e^x - int - 1/2 cosec^2 x/2 e^x dx] + 1/2 int cosec x/2 * e^x  dx`

`= - e^x cot  x/2 - 1/2 int cosec^2  x/2 * e^x dx + 1/2 int cosec^2  x/2 * e^x  dx`

`= - e^x  cot  x/2`

`therefore int_(pi/2)^pi e^x  ((1 - sin x)/(1 + cos x))`dx

`= - [e^x cot  x/2]_(pi/2)^pi`

`= - [pi cot  pi/2 - e^(pi/2) cot  pi/4]`

`= - 0 + e^(pi/2) * 1`

`= e^(pi/2)`