Let f: R → R be such that f(x y)=f(x) f(y), for all x, y ∈ R and f(0) ≠ 0. Let g:[1, ∞) → R be a differentiable function such that x^2 g (x)=∫_1^x(…

Let $\displaystyle f: \mathbf{R} \rightarrow \mathbf{R}$ be such that $\displaystyle f(x y)=f(x) f(y)$, for all $\displaystyle x, y \in \mathbf{R}$ and $\displaystyle f(0) \neq 0$. Let $\displaystyle \mathrm{g}:[1, \infty) \rightarrow \mathbf{R}$ be a differentiable function such that $$x^2 \mathrm{~g}(x)=\int_1^x\left(\mathrm{t}^2 f(\mathrm{t})-\operatorname{tg}(\mathrm{t})\right) d t . $$ Then $\displaystyle \mathrm{g}(2)$ is equal to :