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If int(xe^(x))/(sqrt(1+e^(x)))dx=f(x)sqr...

If `int(xe^(x))/(sqrt(1+e^(x)))dx=f(x)sqrt(1+e^(x))-2logg(x)+C`, then

A

`f(x)=x-1`

B

`g(x)=(sqrt(1+e^(x))-1)/(sqrt(1+e^(x))+1`

C

`g(x)=(sqrt(1+e^(x))+1)/(sqrt(1+e^(x))-1)`

D

`f(x)=2(x-2)`

Text Solution

Verified by Experts

The correct Answer is:
B, D
`int(xe^(x))/(sqrt(1+e^(x)))dx`
`=x(sqrt(1+e^(x)))-2intsqrt(1+e^(x))dx`
`=2xsqrt(1+e^(x))-2int(2t^(2))/(t^(2)-1)dt" "("Putting t"=sqrt(1+e^(x)))`
`=2xsqrt(1+e^(x))-4(t+(1)/(2)ln.(t-1)/(t+1))+C`
`2xsqrt(1+e^(x))-4(sqrt(1+e^(x)))-2ln.(sqrt(1+e^(x))-1)/(sqrt(1+e^(x))+1)+C`
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