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

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.