`int_(pi//3)^(pi//2) (sqrt(1+cosx))/((1-cosx)^(5//2)) dx`

Options

A. `5/2`
B. `3/2`
C. `1/2`
D. `2/5`

Answer

B

Explanation

Let `l = int_(pi//3)^(pi//2)(sqrt(1+cos x))/((1-cos x)^((5)/(2)))dx`
`= int_(pi//3)^(pi//2)(sqrt(1+cos x))/((1-cos x)^((5)/(2)))xx (sqrt(1-cos x))/(sqrt(1-cos x))dx`
`= int_(pi//3)^(pi//2)(sin x)/((1-coc x)^(3))dx`
Put `1-cos x=t rArr sin x dx = dt`
When `x=(pi)/(3), t=(1)/(2)` and `x=(pi)/(2), t=1`
`therefore l=int_(1//2)^(1)(dt)/(t^(3))=[(t^(-2))/(-2)]_(1//2)^(1)=(3)/(2)`.