An improper integral of Type 1 is an integral whose limits of integration include $\infty$ or $-\infty$, or both. Remember that $\infty$ is a process (keep going and never stop), not a number! The integral $$\int_a^\infty f(x) dx \qquad \hbox{means} \qquad \lim_{t \to \infty} \int_a^t f(x) dx,$$and likewise$$\int_{-\infty}^b f(x) dx = \lim_{t \to -\infty} \int_t^b f(x) dx.$$ When an integral runs from $-\infty$ to $\infty$, you have to break the integral into two pieces:$$\int_{-\infty}^\infty f(x) dx = \int_{-\infty}^a f(x) dx + \int_a^\infty f(x) dx,$$where you can pick the break point $a$ to be any number you like. (Zero is often convenient.) To evaluate the limits a $t \to \infty$ or $t \to -\infty$, you sometimes need to use L'Hospital's rule.

An improper integral integral is said to converge if the limit as $t\to\infty$ or $t\to -\infty$ exists, and to diverge if it doesn't. We'll be talking a lot more about convergence and divergence when we get to sequences and series.

The following video explains Type 1 improper integrals and works out a number of examples.