intergal of $|f(x)|/f(x)$

Question

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.

The reverse of the chain rule is integration by substitution. — Chai T. Rex, Sep 24 '17 at 02:41
Integration by substitution is a very poor choice of techniques for solving this integral, just as the chain rule is a very poor choice for taking the derivative of $\frac{\lvert f(x)\rvert}{f(x)}.$ — David K, Sep 24 '17 at 03:04

score 0 · Accepted Answer · answered Sep 24 '17 at 03:07

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 $

intergal of $|f(x)|/f(x)$

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