\begin{definition}[Definition:Integrable Function/Unbounded]
Let $f: \R \to \R$ be a real function.
Let $f$ be unbounded on the open interval $\openint a b$.
Let:
:$f^+$ denote the positive part of $f$
:$f^-$ denote the negative part of $f$.
Let $f^+$ and $-f^-$ both be integrable on $\openint a b$.
Then $f$ is '''integrable on $\openint a b$ and its (definite) integral is understood to be:
:$\ds \int_a^b \map f x \rd x := \int_a^b \map {f^+} x \rd x - \int_a^b \paren {-\map {f^-} x} \rd x$
\end{definition}