If $\int f(x) d x=f(x)$, then $\int\{f(x)\}^2 d x$ is equal to
Answer & explanation
Correct answer: option 1
We have,
$\int f(x) d x=f(x) \Rightarrow \frac{d}{d x}(f(x))=f(x)$
$\Rightarrow \frac{1}{f(x)} d(f(x))=d x$
$\Rightarrow \log (f(x))=x+\log C$
$\Rightarrow f(x)=C e^x$
$\Rightarrow \{f(x)\}^2=C^2 e^{2 x}$
$\Rightarrow \int\{f(x)\}^2 d x=\int C^2 e^{2 x} d x=\frac{C^2 e^{2 x}}{2}=\frac{1}{2}\{f(x)\}^2$