Improper integral
A definite integral with an infinite bound, or an integrand that blows up somewhere in the interval.
An improper integral has either an INFINITE limit of integration (\int_a^{\infty}f(x)\,dx) or an integrand with an INFINITE DISCONTINUITY somewhere in [a,b]. Both are handled by replacing the troublesome bound with a variable (say t), evaluating the resulting proper integral, and taking a limit as t\to\infty or t approaches the discontinuity — the improper integral CONVERGES if that limit exists (is finite) and DIVERGES otherwise.
Examples
\int_1^{\infty}\frac{1}{x^2}\,dx=\lim_{t\to\infty}\int_1^t x^{-2}\,dx=\lim_{t\to\infty}\left(1-\frac1t\right)=1 — converges.
\int_1^{\infty}\frac{1}{x}\,dx diverges — the same limit process gives \lim_{t\to\infty}\ln t=\infty.