Improper integrals are integrals where either:
Here’s an example of the first type of improper integral:
$$ \int_0^\infty e^{-x}\,dx $$
What does this even mean? How can we integrate over an infinite interval?
Well, let’s just see what happens if we take the standard integral $\int_0^b e^{-x}\,dx$ and increase the upper bound to larger and larger numbers, letting it approach infinity:
https://chaddypratt.org/math2419/improper-integral
As you can see, as we let the upper bound $b$ approach infinity, the total area under the curve approaches 1. This means that it makes sense to say, in a sense, that the area under the curve from $x=0$ to $x=\infty$ is 1.
More formally, we can find this improper integral by rewriting it as a limit where the upper bound $b$ approaches infinity:
$$ \int_0^\infty e^{-x}\,dx = \lim_{b\to\infty}\int_0^b e^{-x}\,dx = \lim_{b\to\infty}(-e^{-x})\bigg|0^b = \lim{b\to\infty}[-e^{-b} - (-e^{-0})] =\lim_{b\to\infty}[-e^{-b} + 1] = 1 $$
In general:
$$ \int_a^\infty f(x)\,dx = \lim_{b\to \infty}\int_a^b f(x)\,dx $$
We can do something similar for integrals where the lower bound is $-\infty$.
$$ \int_{-\infty}^b f(x)\,dx = \lim_{a\to -\infty}\int_a^b f(x)\,dx $$
What about integrals where both bounds are infinite? In that case, we can use integral properties to split the integral into two separate improper integrals.
$$ \int_{-\infty}^\infty f(x)\,dx = \int_{-\infty}^cf(x)\,dx + \int_c^\infty f(x)\,dx $$
Here, $c$ can be any real number. Note that both of the integrals on the right-hand side must be convergent for the left-hand integral to be convergent (I will explain what convergent integrals are soon).