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

$$\displaystyle \int_0^{+\infty}e^{-x} \ dx= \lim_{b \to \infty} {\int_0^b e^{-x} \ dx} = \lim_{b \to \infty}{\Big[-e^{-x}\Big]_0^b}=\lim_{b \to \infty}{(1-e^{-b})}=1$$

## Improper Integrals of the second type

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

## Improper Integrals of third type

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