Improper Riemann Integrals

Iniitally Riemann integrals are defined only for bounded functions defined on a set of compact intervals.

Type 1

A function that is not integrable at one endpoint of the interval.

Suppose $f:(a,b]\to\mathbb{R}$ is integrable on $[c,b]\;\forall c\in (a,b)$.

Can be similarly defined on the other endpoint. The above integral converges iff the limit exists and finite. Otherwise diverges.

Type 2

A function defined on unbounded interval (including $\infty$).

Suppose $f:[a,\infty)\to\mathbb{R}$ is integrable on $[a,r] \forall r > a$.

Can be similarly defined on the other endpoint. The above integral converges iff the limit exists and finite. Otherwise diverges.

Type 3

A function that is undefined at finite number of points. The integral can be split into multiple integrals of type 1. Similarly integrals from $-\infty$ to $\infty$ can be defined.

Convergence of improper integrals is similar to the convergence of series.

Absolute Convergence Test

Common Integrals

The above integrals converge iff $p$ is in the integrating (open) interval. Converges to $\frac{1}{p-1}$ in that case.

Both of the above integrals converges. Direct comparison test can be used.

• For the 1st integral, $\sqrt{x^{-1}}$ can be used
• For the 2nd integral, $x^{-2}$ can be used