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

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

A

`log_(e)(1+sqrt(2))`

B

`log_(e)(1+sqrt(2))-(pi)/(4)`

C

`log_(e)(1+sqrt(2))+(pi)/(4)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
b
We have , `f(x) = int(x^(2))/((x^(2)+1){1+sqrt(1+x^(2))})dx`
`rArrf(x) = int((sqrt(1+x^(2))-1))/((x^(2)+1))dx=int(1)/(sqrt(x^(2)+1))dx=int(1)/(x^(2)+1)dx`
`rArrf(x) = log_(e)(x+sqrt(x^(2)+1))-tan^(-1)x+C`
Now , `f(0) =0= rArr0=log_(e)1-tan^(-1)0+CrArrC=0`
`thereforef(x) =log_(e)(x+sqrt(x^(2)+1))-tan^(-1)x`
`rArr f(1)=log_(e)(sqrt(2)+1)-(pi)/(4)`
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