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 $[c,b]\;\forall c\in (a,b)$.

$$\int_a^b f = \lim_{\epsilon \to 0}\; \int_{a+\epsilon}^b f$$

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 $[a,r] \forall r > a$.

$$\int_a^\infty f = \lim_{r \to \infty}\; \int_a^r f$$

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.

Note

The integral can be split into multiple integrals only when all those integrals exist.

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

Absolute convergence test

$$\int_a^b \lvert f \rvert\;\text{converges} \implies \int_a^b f \;\text{converges}$$

Common integrals

$$\int_0^1 \frac{1}{x^p}\,\text{d}x \;\;\; \;\;\; \;\;\; \int_1^\infty \frac{1}{x^p}\,\text{d}x$$

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

$$\int_0^1 \frac{\sin^2 x}{x^2}\,\text{d}x \;\;\; \;\;\; \;\;\; \int_1^\infty \frac{\sin^2 x}{x^2}\,\text{d}x$$

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