If $\int f(x) d x=f(x)$, then $\int\{f(x)\}^2 d x$ is equal to

$\frac{1}{2}\{f(x)\}^2$

We have,

$\int f(x) d x=f(x) \Rightarrow \frac{d}{d x}(f(x))=f(x)$

$\Rightarrow \frac{1}{f(x)} d(f(x))=d x$

$\Rightarrow \log (f(x))=x+\log C$

$\Rightarrow f(x)=C e^x$

$\Rightarrow \{f(x)\}^2=C^2 e^{2 x}$

$\Rightarrow \int\{f(x)\}^2 d x=\int C^2 e^{2 x} d x=\frac{C^2 e^{2 x}}{2}=\frac{1}{2}\{f(x)\}^2$