Improper Integrals

Definition

For integrals like ${\displaystyle {\int }_a^{b}f(x)dx}$, if the upper limit or lower limit (or both) is infinite, or if the integrand is unbounded near the endpoint, we call it a improper integral.

Convergence or Divergence

For improper integrals with infinite limits of integration, if the value of the integral is approaching a real number when the upper(lower) limit is arbitrarily big(small), we say it is converge. If not, it is diverge.

For those with infinite integrands at endpoints, if the value of the integral is approaching a real number when the upper(lower) limit is arbitrarily close to the endpoint, we say it is converge. If not, it is diverge.

For improper integrals with both infinite limits and infinite integrands, both the condition above should be satisfied to be converge.

Calculate the Improper Integral

The calculation is based on the condition that the improper integral is converge.

To find the value, we need to find the limit of the integral:

1. First we need to do the integration to find the antiderivative.
2. Then is to replace the upper and lower limit with variables.
3. Divide the integral interval by a chosen point ${\displaystyle c}$ if both upper and lower limits are infinite.
4. Make the antiderivative to the definite integral.
5. Find each limit and add them together.

For improper integrals with infinite integrands, find the limit at the infinite endpoint.