Is there a way to take the integral of $\frac{\lvert f(x)\rvert}{f(x)}$ in terms of x?
I'm a little new to integrals, but I've had experience with derivatives, so I was thinking there was a chain rule for integrals, but I couldn't find an effective way.
Let $ g(x) = \frac{|f(x)|}{f(x)} $
Then g(x) can be defined by breaking it into three intervals as follows:
g(x) = 1, if f(x) > 0
Not defined, if f(x) = 0
-1, if f(x) < 0
Integrating the function interval-wise:
x, if f(x) > 0
Not defined, if f(x) = 0
-x, if f(x) < 0
The integral can be written as one single term too, i.e. $ x \cdot \frac{|f(x)|}{f(x)} + C $