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 R-integrable on every finite interval.

• If $f:[a,+\mathrm{\infty })\to \mathbb{R}$, we say

converges if the finite limit exists:

• If $f:(-\mathrm{\infty },b]\to \mathbb{R}$, define

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

Type B: unbounded integrand at an endpoint

Assume $f$ is Riemann integrable on each truncated interval $[a+\epsilon ,b]$ for $0<\epsilon <{\epsilon }_0$ for certain ${\epsilon }_0>0$.

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

if the limit above exists and is finite. We say that ${\int }_a^{b}f(x)dx$ is convergent.

• If $f:[a,b)\to \mathbb{R}$ and $f$ may blow up at $b$, we proceed analogously.

• Let $f:(a,b)\to \mathbb{R}$ be a Riemann-integrable (R-integrable) function in every interval $[c,d]\subset (a,b)$. We say that ${\int }_a^{b}f(x)dx$ is convergent if, for some $c\in \mathbb{R}$, the integrals ${\int }_a^{c}f(x)dx$ and ${\int }_c^{b}f(x)dx$ are convergent; in such case, we shall set: $$\int_{a}^{b} f(x)dx = \int_{a}^{c} f(x)dx + \int_{c}^{b} f(x)dx$$

Type C: unbounded at an interior point

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

Cauchy criterion (useful when primitives are hard)

Suppose ${\int }_a^{b}f$ is improper at $b$ (either $b=+\mathrm{\infty }$ or $\underset{x\to b^{-}}{lim}f(x)=+\mathrm{\infty }$). Then the improper integral converges iff:

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

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

Related

• Riemann integral
• p-test for improper integrals