Triangle Inequality, Integrals

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The discussion centers on the relationship between the integral of a function and the integral of its absolute value, questioning whether the inequality |∫ f(x) dx| < ∫ |f(x)| dx holds in general. It is confirmed that this inequality is indeed true, as it follows from the properties of absolute values. The reasoning is based on the fact that -|f(x)| ≤ f(x) ≤ |f(x)|, which supports the validity of the inequality. The term "Triangle Inequality" is mentioned as potentially relevant to this concept. Overall, the discussion affirms the general truth of the stated inequality in integrals.
psholtz
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Is it true in general that:

|\int f(x)dx| &lt; \int |f(x)|dx

Not sure if "Triangle Inequality" is the right word for that, but that seems to be what's involved.
 
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It is true in general simply because -|f(x)|≤f(x)≤|f(x)|.
 

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