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int(pi/4)^(pi/2) e^x(logsinx+cotx)dx...

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

A

` e^(pi//4) log2 `

B

` -e^(pi//4) log2 `

C

` 1/2 e^(pi//4) log2`

D

` -1/2 e^(pi//4) log 2 `

Text Solution

Verified by Experts

The correct Answer is:
C
Let `l=int_(pi//4)^(pi//2)e^(x)` (log sin x + cot x)dx
`rArr l = int_(pi//4)^(pi//2)e^(x)` log sin x dx `+ int_(pi//4)^(pi//2)e^(x)` cot x dx
`= int_(pi//4)^(pi//2)e^(x)` log sin x dx `+ [e^(x) log sin x]_(pi//4)^(pi//2)-int_(pi//4)^(pi//2)e^(x)` log sin x dx
`= e^(pi//2)log sin.(pi)/(2)-e^(pi//4)log sin.(pi)/(4)`
`= -e^(pi//4)log((1)/(sqrt(2)))`
`=(1)/(2)e^(pi//4)log 2`
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