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I've just noticed that integrals of expressions like this are usually assumed to be equal to the integral of f(x) without the absolute value. I'd like to know how that's possible.

Is weird for me to consider those expressions; specially because of what Minkowski's inequality says about it:

[tex] \int{|f(x)|\,dx} \leq \left| \;{\int{f(x)dx}}\; \right| [/tex]

In general, I would expect the integral without the absolute value to be different from the "absolute-valued" one.

Check, for example:

[tex] \int_\pi^{3\pi/2} |\sin(x)| \, dx [/tex] shouldn't give a negative value as a result, however, that's what WolframAlpha gives me. [Just like if it were ignoring the absolute value and just integrating sin(x) ]