Let f : [0,2] `rarr` R be a function which is continuous on [0 , 2] and is differentiable on (0 , 2) with f(0) = 1. Let
F(x) =`int_0^(x^2) f(sqrtt)dt`
for `x in [0,2]`. If F'(x) = f'(x) for all `x in (0,2)`, then F(2) equals

Options

A. `e^(2)-1`
B. `e^(4)-1`
C. `e-1`
D. `e^(4)`

Answer

B

Explanation

`F(x)=int_(0)^(x^(2))f(sqrt(t))dt`
`F(0)=0`
`F'(x)=2xf(x)=f'(x)`
`implies int(f'(x))/(f(x))dx=int2xdx`
`implieslog_(e)f(x)=x^(2)+c`
`impliesf(x)=e^(x^(2)+c) ( :' f(0)=1)`
`impliesf(x)=e^(x^(2))`
`F(x)=int_(0)^(x^(2))e^(t)dt`
`impliesF(x)=e^(x^(2))-1 ( :'F(0)=0)`
`impliesF(2)=e^(4)-1`