Improper integrals

An improper integral is an integral that has a vertical asymptote in the integration interval, or an integral with not bounded integration interval.

In this type of integrals, we will calculate the integral in an interval reduced with a parameter and then we will to do the limit of the result.

An improper integral might not converge, meaning that the result might be infinite. There are 3 types of improper integrals.

Improper Integrals of the first type

Integrals of the type $\int_{- \infty}^{b} f (x) d x$ or $\int_{a}^{+ \infty} f (x) d x$ . These integrals will be solved in the following way: $\begin{array}{l} \int_{- \infty}^{b} f (x) d x = lim_{c \to + \infty} \int_{c}^{b} f (x) d x \\ \int_{a}^{+ \infty} f (x) d x = lim x \to \infty \int_{a}^{c} f (x) d x \end{array}$

Example

$\int_{0}^{+ \infty} e^{- x} d x = lim_{b \to \infty} \int_{0}^{b} e^{- x} d x = lim_{b \to \infty} [- e^{- x}]_{0}^{b} = lim_{b \to \infty} (1 - e^{- b}) = 1$

Improper Integrals of the second type

These integrals are $\int_{a}^{b} f (x) d x$ where $f (x)$ has a discontinous asymptote in the integration interval. If this discontinuity is in the interval $c \in [a, b]$ , then $\int_{a}^{b} f (x) d x = lim_{d \to c^{-}} \int_{a}^{d} f (x) d x + lim_{e \to c^{+}} \int_{e}^{b} f (x) d x$

Improper Integrals of third type

They are a mixture of the first and the second type.