= Absolutely convergent improper integral
{title2=$\int_I|f(x)|\,dx<\infty$}
= Absolutely convergent integral
{synonym}
An <improper integral> whose absolute-value integral is finite. The <triangle inequality for an integral> bounds each truncated tail by the corresponding tail of $|f|$, so the <Cauchy criterion for an improper integral> gives ordinary convergence. Absolute convergence is stronger than a <Cauchy principal value>, which may exist only because two divergent sides cancel. The continuous sine-product integrand $\sin(ax)\sin(bx)/x^2$ has a removable value $ab$ at zero and an integrable $1/x^2$ tail; its auxiliary complex contour integrand can nevertheless require a principal value.
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