Integral of the absolute value of an unknown function

Question

I was just wondering if there is a general way (method/formula) to antidifferentiate the absolute value of a function. As in, is there any way to find $\int|f(x)|dx $ given that I already know what $\int f(x)dx$ is? I am referring to the indefinite integral.

Answer 1

$$\int|f()| = \frac{|f(x)|}{()}\cdot ∫f(x)dx + c$$

Proof:

$$\frac{d}{dx}\left(\frac{|f(x)|}{f(x)}\cdot\int f(x)dx\right)$$

$$= \frac{d}{dx}(\text{sgn}(f(x)))\cdot\int f() + \frac{d}{dx}\left(\int f()\right)\cdot \frac{|f()|}{f()}$$ as $$\text{sgn}(f(x))=\frac{|f(x)|}{f(x)}$$

$$= 0 + f(x)\cdot \frac{|f()|}{f()}$$ as $$\frac{d}{dx}(\text{sgn}(f(x)))=0$$ for all $f(x) \neq 0$

$$= |f(x)|$$

Therefore, $\frac{|f(x)|}{f(x)}\cdot\int f(x)dx$ is a solution to $\int|f(x)|dx$ for all $f(x) \neq 0$ so the general solution must be $\frac{|f(x)|}{()}\cdot\int f(x)dx + c$ for all $f(x) \neq 0$