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# Improper Intervals: Infinite Intervals of Integration and Infinite Discontinuities for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus

### Infinite Intervals of Integration

Improper integrals are integrals with infinite intervals of integration or infinite discontinuities within the interval of integration. For infinite intervals of integration, If the limit exists, the integral converges. for some value c.

#### Example 1

Evaluate

so the integral diverges.

#### Example 2

Evaluate

Since both limits of integration are infinite, consider the sum of the improper Integrals This sum is the sum of the limits Since the limit exists, the integral converges and

### Infinite Discontinuities

If the function has an infinite discontinuity at one of the limits of integration, then If an infinite discontinuity occurs at x =c within the interval of integration (a, b), then the integral can be broken into sections at the discontinuity and the sum of the two improper integrals can be found.

#### Example

Evaluate

Since has an infinite discontinuity at the integral is improper.

Evaluate Since the limit exists,

Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus

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