Suppose I'm given $f(x)$ and $f'(x)$ in closed form. Further suppose that $f(x)$ is continuously differentiable. I'm trying to find a nonconstant continuously differentiable function $g$ whose image is $\mathbb{R}$ that can be expressed in closed form such that
$$ \int g(f'(x)) dx$$
can be expressed in closed-form as a function of $f(x)$ or $f'(x)$ or both.
and $g(f'(x)) > 0$.
I've tried $g(x) = e^x$ and $g(x) = x^2$. However, using WolframAlpha, I find that the integral cannot be expressed in closed-form. Can I get a hint on this? Is this even possible?
