An improper integral is an integral that either has infinite limits of integration or an integrand that approaches infinity at one or more points in the interval of integration. Improper integrals can be evaluated using the same techniques as definite integrals, but require additional analysis to determine if they converge or diverge.

To determine if an improper integral converges or diverges, it is necessary to analyze the behavior of the integrand as the limits of integration approach infinity or as the integrand approaches infinity within the interval of integration. If the integral is finite, it converges. If the integral is infinite or does not exist, it diverges.

There are several convergence tests that can be applied to improper integrals, including the comparison test, the limit comparison test, the integral test, and the ratio test. These tests involve comparing the integrand to a known function or series with known convergence behavior.

Improper integrals can be classified into two types: Type 1 and Type 2.

Type 1 improper integrals have infinite limits of integration. They can be evaluated by taking the limit as the upper or lower limit approaches infinity, or by breaking up the integral into smaller intervals and evaluating each interval separately.

For example, the improper integral $\int_0^{\infty} \frac{1}{x^2} dx$ is a Type 1 improper integral because the upper limit of integration is infinite. Evaluating this integral involves taking the limit as the upper limit approaches infinity:

$\int_0^{\infty} \frac{1}{x^2} dx = \lim_{b \to \infty} \int_0^b \frac{1}{x^2} dx = \lim_{b \to \infty}[-\frac{1}{x}]_0^b = \lim_{b \to \infty}[-\frac{1}{b} + \frac{1}{0}]$

Since the limit of the integral evaluates to a finite value, the integral converges.

Type 2 improper integrals have integrands that approach infinity at one or more points in the interval of integration. They can be evaluated by splitting the interval of integration at the point(s) of infinity and analyzing the behavior of the integrand on each interval.

For example, the improper integral $\int_0^1 \frac{1}{\sqrt{x}} dx$ is a Type 2 improper integral because the integrand approaches infinity as $x$ approaches zero. Evaluating this integral involves splitting the interval of integration at $x=0$ and analyzing the behavior of the integrand on each interval:

$\int_0^1 \frac{1}{\sqrt{x}} dx = \lim_{a \to 0^+} \int_a^1 \frac{1}{\sqrt{x}} dx = \lim_{a \to 0^+} [2\sqrt{x}]_a^1 = 2 - \lim_{a \to 0^+} 2\sqrt{a}$

Since the limit of the integral evaluates to a finite value, the integral converges.

Improper integrals have a variety of applications in mathematics and the physical sciences, including in the calculation of areas, volumes, and probabilities. They are also used in the solution of differential equations and the evaluation of Laplace transforms.

• Laplace Transform
• Gamma Function
• p-series