Improper integrals describe integrals which violate the usual assumptions of Riemann sums, that is having a closed interval and a bounded integrand (the function that is being integrated). Improper integrals take one of the following 4 terms:

1. , where is some constant defined real number.
2. , where is some constant defined real number.
4. , where is undefined or discontinuous at some point on the interval .

Vergence

If the limit we use to calculate the improper integral does not exists or is not finite we can say the improper integral diverges otherwise if it can be determined to be a finite value we say it converges. If any of the limits in an equation do no exists or are infinite, then the whole equation is said to diverge.

Integrating

Non-Continuous or Undefined Integral

If has a discontinuity at some , we define

Origin

If exists for all , the we can define

If exists for all , the we can define

First we apply the sum rule to the integral splitting it into two parts, one on the left side of the discontinuity and the second on the right side

Where is the point of discontinuity.

Then using the rules we defined earlier we can derive the full equation.

Examples

Example 1 - Converging

Find .

There is a discontinuity at so we set .

Example 2 - Diverging

Does converge or diverge?

Since the limit does not exists we can say this improper integral diverges

Infinite Intervals

An improper integral can also arise when one or both of the limits of integration are infinite. In these cases, we evaluate the integral by taking the limit as the interval extends to infinity.

Integrals with Infinite Upper Limit

For an integral of the form , we define it as follows:

Integrals with Infinite Lower Limit

For an integral where the lower limit is infinite, , we define it as:

Double Infinite Intervals

For integrals where both limits are infinite, such as , we first split the integral using the sum rule:

Where is some arbitrarily chosen point, that will make these integrals easy to calculate.

Then applying our rules for integrals with infinite upper limits and infinite lower limits.