Let $f:(0, \infty) \rightarrow \mathbb{R}$ and $F(x)=\int_{0}^{x} f(t) d t$. If $F(x)= x^{2}(1+x)$,

Question

Let $f:(0, \infty) ightarrow \mathbb{R}$ and $F(x)=\int_{0}^{x} f(t) d t$. If $F(x)= x^{2}(1+x)$, then $f(2)$ is equal to\

Professor PC Thomas Classes · Answer

(d): Given, $F(x)=x^{2}(1+x) ; F(x)=\int_{0}^{x} f(t) d t$\ $\Rightarrow \frac{d}{d x}\left[x^{2}(1+x)ight]=\frac{d}{d x} \int_{0}^{x} f(t) d t$\ $\Rightarrow \quad 2 x+3 x^{2}=f(x)$\ (Using Leibnitz theorem)\ Hence, $f(2)=2(2)+3(2)^{2}=4+12=16$\