Mathematics - Definite Integration Question with Solution | TestHub

Differentiation gives $3f^{2}f\prime =x{f}^2$, hence $f\prime =x/3$ where applicable; with $f(0)=0,f=x^2/6$. Thus ${\int }_0^{1}f=1/18$. Its range is $[0,\infty )$, not all $\mathbb{R};x^2=e^x$ has no real solution, and $x^2=3x^2+2x+2$ has no real solution. Therefore (A), (C), (D) are incorrect.