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Let f(x)=int x^2/((1+x^2)(1+sqrt(1+x^2))...

Let `f(x)=int x^2/((1+x^2)(1+sqrt(1+x^2)))dx` and `f(0)=0` then `f(1)` is

A

`log(1+sqrt2)`

B

`log(1+sqrt2)-(pi)/(4)`

C

`log(1+sqrt2)+pi//4`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
On putting `x=tan theta rArr dx=sec^(2) theta d theta`, we get
`f(x)=int(tan^(2) theta.sec^(2)theta)/(sec^(2) theta(1+sec theta))d theta`
`=int(sec^(2) theta-1)/(1+sec theta)d theta=int(sec theta-1) d theta`
`=log (sec theta tan theta)- theta+C`
`rArr" "f(x)=log(sqrt(1+x^(2))+x)-tan^(-1)x+C`
At `x=0," "f(0)=log(1+0)-0+C rArr C=0`
At `x=1," "f(1)=log(1+sqrt2)-(pi)/(4)`
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