Let `f:[0,2]vecR` be a function which is continuous on [0,2] and is differentiable on (0,2) with `f(0)=1` `L e t :F(x)=int_0^(x^2)f(sqrt(t))dtforx in [0,2]dotIfF^(prime)(x)=f^(prime)(x)` . for all `x in (0,2),` then `F(2)` equals `e^2-1` (b) `e^4-1` `e-1` (d) `e^4`

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`