Mathematics - Definite Integration Question with Solution | TestHub

From given integral equation, $f(0)=0$. Also, differentiating the given integral equation w.r.t. $x$, we get $$\begin{array}{c}{f}^{\prime }(x)=f(x)\\ \text{ If }f(x)\ne 0\Rightarrow \frac{f^{\prime }(x)}{f(x)}=1\\ \Rightarrow \log f(x)=x+C\Rightarrow f(x)=e^C{e}^x\\ \because f(0)=0\Rightarrow e^C=0,\text{ a contradiction }\\ \therefore f(x)=0,\forall x\in R\Rightarrow f(\ln 5)=0\end{array}$$