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If f'(x)=(dx)/((1+x^(2))^(3//2)) and f(0...

If `f'(x)=(dx)/((1+x^(2))^(3//2)) and f(0)=0.` then f(1) is equal to

A

`sqrt2`

B

`-(1)/(sqrt2`

C

`(1)/(sqrt2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
`f'(x)=(dx)/((1+x^(2))^(3//2))`
On integrating both sides, we get
`f(x)=int(dx)/((1+x^(2))^(3//2))+C`
Put ` x= tan theta rArr dx=sec^(2) theta d theta`
`therefore" "f(x)=int(sec^(2)theta)/(sec^(3)theta)d theta+C = int cos theta d theta +C`
`rArr" "f(x)=sin theta +C`
`rArr" "f(x)=(x)/(sqrt(1+x^(2)))+C`
`rArr" "f(0)=0+C rArr C=0`
`therefore" "f(x)=(x)/(sqrt(1+x^(2))) rArr f(1)=(1)/(sqrt2)`
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