Improper integral

The (Riemann) integral is originally defined for bounded functions on bounded intervals. An improper integral extends the notion by defining it as a limit of proper integrals.

Type A: unbounded interval

Let $f$ be Riemann integrable on every finite interval.

If $f : [a, + \infty) \to R$ , we say

\int_{a}^{+ \infty} f (x) d x

converges if the finite limit exists:

\int_{a}^{+ \infty} f (x) d x := lim_{b \to + \infty} \int_{a}^{b} f (x) d x .

If $f : (- \infty, b] \to R$ , define

\int_{- \infty}^{b} f (x) d x := lim_{a \to - \infty} \int_{a}^{b} f (x) d x .

If $f : R \to R$ and both one-sided improper integrals converge for some $c \in R$ , define

\int_{- \infty}^{+ \infty} f (x) d x := \int_{- \infty}^{c} f (x) d x + \int_{c}^{+ \infty} f (x) d x .

Type B: unbounded integrand at an endpoint

Assume $f$ is Riemann integrable on each truncated interval.

If $f : (a, b] \to R$ and $f$ may blow up at $a$ , define

\int_{a}^{b} f (x) d x := lim_{ε \to 0^{+}} \int_{a + ε}^{b} f (x) d x .

If $f : [a, b) \to R$ and $f$ may blow up at $b$ , define

\int_{a}^{b} f (x) d x := lim_{ε \to 0^{+}} \int_{a}^{b - ε} f (x) d x .

Type C: unbounded at an interior point

If $f : (a, b) \to R$ has a singularity at $c \in (a, b)$ , define convergence by splitting:

\int_{a}^{b} f (x) d x converges if both \int_{a}^{c} f (x) d x and \int_{c}^{b} f (x) d x converge.

Cauchy criterion (useful when primitives are hard)

Suppose $\int_{a}^{b} f$ is improper at $b$ (either $b = + \infty$ or $lim_{x \to b^{-}} f (x) = + \infty$ ). Then the improper integral converges iff:

For every $ε > 0$ there exists $c \in (a, b)$ such that

| \int_{t_{1}}^{t_{2}} f (x) d x | < ε whenever c < t_{1} < t_{2} < b .

(A similar statement holds when the integral is improper at $a$ .)

Riemann integral