Summary of Improper Integrals

Essential Concepts

  • Integrals of functions over infinite intervals are defined in terms of limits.
  • Integrals of functions over an interval for which the function has a discontinuity at an endpoint may be defined in terms of limits.
  • The convergence or divergence of an improper integral may be determined by comparing it with the value of an improper integral for which the convergence or divergence is known.

Key Equations

  • Improper integrals

    [latex]\begin{array}{c}{\displaystyle\int }_{a}^{+\infty }f\left(x\right)dx=\underset{t\to \text{+}\infty }{\text{lim}}{\displaystyle\int }_{a}^{t}f\left(x\right)dx\hfill \\ {\displaystyle\int }_{\text{-}\infty }^{b}f\left(x\right)dx=\underset{t\to \text{-}\infty }{\text{lim}}{\displaystyle\int }_{t}^{b}f\left(x\right)dx\hfill \\ {\displaystyle\int }_{\text{-}\infty }^{+\infty }f\left(x\right)dx={\displaystyle\int }_{\text{-}\infty }^{0}f\left(x\right)dx+{\displaystyle\int }_{0}^{+\infty }f\left(x\right)dx\hfill \end{array}[/latex]


improper integral
an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges