An [[Improper Integrals|Improper Integral]] of Type 2 is an [[Integration|Integral]] over the range of a function with a [[Continuous|Discontinuous]] point.
If $f$ is [[Continuous]] on $[a, b)$ but discontinuous at $b$.
$\huge
\begin{align}
\int_{a{}}^{b} f(x)\d x &=
\lim_{to\to b^{-}} \int_{a}^{t}f(x)\d x
\end{align}
$
<br>
If $f$ is [[Continuous]] on $(a, b]$ but discontinuous at $a$.
$\huge
\begin{align}
\int_{a{}}^{b} f(x)\d x &=
\lim_{to\to a^{+}} \int_{t}^{b}f(x)\d x
\end{align}
$
If $f$ has a discontinuity at $c$, where $a < c<b$, and $f$ is convergent on $[a,c)$ and $(c, b]$,
$\huge
\begin{align}
\int_{a{}}^{b} f(x)\d x &=
\underbrace{
\int_{a{}}^{c} f(x)\d x +
\int_{c{}}^{b} f(x)\d x }_{\color{lightgray}\text{Improper Integrals}}
\end{align}
$