Improper integral

The premise of improper integrals is the idea that sometimes our bounds for definite integrals extend infinitely, either on the $x$-axis or vertically with an asymptote.

We define, with a function $f(x)$ that is continuous on the interval $[a,\infty ]$:

$${\int }_a^{\infty }f(x)dx=\underset{t\to \infty }{\lim }{\int }_a^{t}f(x)dx$$

If this limit exists, the integral will converge. Otherwise it will diverge.

See also

• Principal value integral, an extension of the notion of improper integrals