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The value of the integral int0^(log5)(e^...

The value of the integral `int_0^(log5)(e^xsqrt(e^x-1))/(e^x+3)dx`

A

`3+2pi`

B

`4-pi`

C

`2+pi`

D

`2-pi`

Text Solution

Verified by Experts

The correct Answer is:
B
Put `e^(x)-1=t^(2)`
`I=2.int_(0)^(2)(t^(2))/(t^(2)+4)dt=2(int_(0)^(2)dt - 4int_(0)^(2)(dt)/(t^(2)+4))`
`= 2.[(t-2tan^(-1)((t)/(2)))_(0)^(2)]=2.[2-2((pi)/(4))]`
`= 4 - pi`
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