Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& {\int }_0^{x}tf(t)dt=x^{2}f(x)\\ & \Rightarrow xf(x)=2xf(x)+x^2{f}^{\prime }(x)\\ & \Rightarrow x^2\cdot f^{\prime }(x)=-xf(x)\\ & \Rightarrow \frac{f^{\prime }(x)}{f(x)}=-\frac{1}{x}\\ & \\ & \int \frac{f^{\prime }(x)}{f(x)dx}=-\int \frac{1}{x}dx\\ & \Rightarrow \ln f(x)=-\ln x+C\\ & \\ & f(2)=3\Rightarrow \ln 3=-\ln 2+C\\ & C=\ln 6\\ & \Rightarrow f(x)=\frac{6}{x}\Rightarrow f(6)=1\end{array}$