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

Options

A. `sqrt(2)`
B. `- ( 1)/( sqrt(2))`
C. `(1)/(sqrt(2))`
D. None of these

Explanation

`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))`