Answer
\[f(x)=\frac{x}{2}\]
Work Step by Step
It is given that $f(x)$ is never zero and $f(x)$ is differentiable and
\[\int_{0}^{x}f(t)\:dt=[f(x)]^2\;\;\;...(1)\]
For all $x$
Differentiating (1) with respect to $x$
\[\frac{d}{dx}\int_{0}^{x}f(t)\:dt=2[f(x)]\cdot f'(x)\;\;\;...(2)\]
We will use the formula
\[\frac{d}{dx}\int_{f(x)}^{g(x)}F(t)\:dt=F(g(x))\cdot g'(x)-F(f(x))\cdot f'(x)\;\;\;...(3)\]
Using (3) in (2)
\[f(x)=2f(x)\cdot f'(x)\]
\[\Rightarrow f(x)[2f'(x)-1]=0\]
It is given that $f(x)$ is never zero so
\[2f'(x)-1=0\]
\[\Rightarrow \frac{df}{dx}=\frac{1}{2}\]
\[\Rightarrow f(x)=\int \frac{1}{2}dx\]
\[\Rightarrow f(x)=\frac{x}{2}\]
Hence \[f(x)=\frac{x}{2}\]
