`int_(pi//4)^(pi//2)e^(x)(logsinx+cotx)dx` का मान है

Options

A. `e^(pi//4)log2`
B. `-e^(pi//4)log2`
C. `(1)/(2)e^(pi//4)log2`
D. `-(1)/(2)e^(pi//4)log2`

Answer

D

Explanation

`I=int_(pi//4)^(pi//2)e^(x)(logsinx+cotx)dx`
`=[e^(x)logsinx]_(pi//4)^(pi//2)-int_(pi//4)^(pi//2)cotx dx+int_(pi//4)^(pi//2)e^(x)cotxdx`
`=[e^(x)logsinx]_(pi//4)^(pi//2)`
`=[e^(pi//2)log1-e^(pi//4)logsqrt(2)]`
`=-(1)/(2)e^(pi//4)log2`