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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

`sqrt(2)`

B

`- ( 1)/( sqrt(2))`

C

`(1)/(sqrt(2))`

D

None of these

Text Solution

Verified by Experts

`f'(x) = (1)/( (1 + x^(2))^(3//2))`
On initegrating both sides, `f(x) = int (dx)/((1_ x^(2))^(3//2)) + c`
Put`x = tan theta rArr dx = sec^(2) theta d theta :. f(x) =( 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`
`:. f(x) = ( x)/(sqrt(1 +x^(2))) rArr f(1) =( 1)/( sqrt(2))`
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