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Question
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Solution
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)`
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