Let $f: R \rightarrow R$ be a continuous function, which satisfies $f(x)=\int_0^x f(t) d t$. Then, the value…
Let $f: R \rightarrow R$ be a continuous function, which satisfies $f(x)=\int_0^x f(t) d t$. Then, the value of $f(\ln 5)$ is
Solution
From given integral equation, $f(0)=0$. Also, differentiating the given integral equation w.r.t. $x$, we get
$
\begin{gathered}
f^{\prime}(x)=f(x) \\
\text { If } \quad f(x) \neq 0 \Rightarrow \frac{f^{\prime}(x)}{f(x)}=1 \\
\Rightarrow \quad \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{gathered}
$