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$$\int |f(x)|\, dx = \int f(x) \, dx\cdot\frac{1}{f(x)}\cdot|f(x)|$$ with $|x|$ is the absolute value of $x$.

The equation is nothing new, I guess?


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As long as $f(x) \neq 0$ you have $$f(x)\cdot\frac{1}{f(x)} = 1$$ Multiplying both sides by $|f(x)|$ and integrating yields your identity, which can even be defined as long as $f(x) = 0$ doesn't happen at too many points (integration is very forgiving like that). I can't tell if your result has been explicitly stated before, but there's nothing revolutionary about it.

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