A definite integral is improper when its interval is unbounded or when the integrand becomes unbounded within the interval. Its value is defined through a limit of ordinary proper integrals.

Infinite intervals

For example,

$$\int_a^{\infty}f(x),dx$$

means

$$\lim_{b\to\infty}\int_a^b f(x),dx,$$

provided the limit exists and is finite.

Similarly,

$$\int_{-\infty}^{\infty}f(x),dx$$

is normally split at a finite point and both resulting improper integrals must converge.

Unbounded integrands

If $f(x)$ becomes unbounded as $x\to a^+$, then

$$\int_a^b f(x),dx =\lim_{c\to a^+}\int_c^b f(x),dx$$

when that limit exists.

Convergence

An improper integral either converges to a finite value or diverges. Formal antiderivative manipulation is not enough; the defining limits must converge.

For example,

$$\int_1^{\infty}\frac1{x^2},dx=1,$$

while

$$\int_1^{\infty}\frac1x,dx$$

diverges.

Improper integrals extend accumulation to infinite domains and singular functions and appear naturally in probability densities and integral transforms.