Find, in terms of $f(x)$, $\int\lvert f'(x)\rvert dx$

Question

Find, in terms of $f(x)$, $$\int\lvert f'(x)\rvert dx$$

I don't know if this question has an answer, it just caught my fancy and I'm wondering if it's possible.

I know that $$\frac{d}{dx}\lvert f(x)\rvert=\frac{f(x)f'(x)}{\lvert f(x)\rvert}$$

Do you think that could be useful?

Thanks for your help.

Answer 1

Assuming, for simplicity, that $f'$ has a finite number of zeros $z_1, \cdots, z_n$, and denoting $$ I_0=(-\infty, z_1), I_1=(z_1, z_2), \cdots, I_n = (z_n, +\infty) $$

$|f'(x)|$ has a primitive defined on each $I_k$ by $ F(x) = \textrm{sgn}(f'(x)) f(x) + C_k$. Note that there is not a single integration constant, there can be a different constant on each interval.