Let $f:(0, \infty) \in R$ and $F(x)=\int\limits_0^x f(t) d t$. If $F\left(x^2\right)=x^2(1+x)$, then $f(4)$ equals

4

We have,

$F(x)=\int\limits_0^x f(t) d t \Rightarrow F'(x)=f(x) \Rightarrow f(4)=F'(4)$     .......(i)

Now,

$F\left(x^2\right)=x^2(1+x)=x^3+x^2$

$\Rightarrow F'\left(x^2\right) . 2 x=3 x^2+2 x$

$\Rightarrow F'\left(x^2\right)=\frac{3 x+2}{2}$

$\Rightarrow F'(4)=\frac{3 \times 2+2}{2}=4 \Rightarrow f(4)=4$              [Using (i)]