Type 1 - Improper Integrals with Infinite Intervals of
Integration

An improper integral of type 1 is an integral whose interval of integration is infinite.
This means the limits of
integration include $\infty$ or $-\infty$ or both.
Remember that $\infty$ is a process (keep going and never stop),
not a number. Therefore, we cannot use $\infty$ as an actual
limit of integration as in the FTC II.
We make the limit of integration a number, and take the limit as
that goes to infinity or negative infinity. We define

$\displaystyle\int_{-\infty}^\infty f(x)\, dx =
\int_{-\infty}^a f(x) \,dx + \int_a^\infty f(x) \,dx,$
where we can choose any number for the break point
$a$. (Zero is often convenient.)

On Convergence

$\displaystyle\int_a^\infty f(x)\,dx$ converges
if $\displaystyle\lim_{t\to\infty}\int_a^t f(x)\,dx$ exists, and diverges if the limit doesn't
exist, including when it is infinite.
$\displaystyle\int_{-\infty}^b f(x)\,dx$ converges
if $\displaystyle\lim_{t \to -\infty} \int_t^b f(x)\, dx$ exists,
and diverges if the limit
doesn't exist, including when it is infinite.
$\displaystyle\int_{-\infty}^\infty f(x)\,dx$ converges if both $\displaystyle\int_a^\infty
f(x)\,dx$ and $\displaystyle\int_{-\infty}^b f(x)\,dx$ converge.

We'll be talking a lot more about
convergence and divergence when we get to sequences and
series.

The following video explains improper integrals with infinite
intervals of integration (type 1) and works out a number of
examples.