Comparison Test for Improper Integrals

In a Nut Shell: Sometimes improper integrals involve complicated expressions that

cannot be integrated. Yet, one still wants to determine if the improper integral

converges or not. The “comparison test” provides a way to evaluate such integrals.

Strategy: Consider the original improper integral:

∞

I = ∫ f(x) dx

a

where f(x) is a complicated function. The strategy is to find another integral with

a simpler function, g(x), that you can evaluate

∞

I = ∫ g(x) dx

a

where f(x) ≥ g(x) ≥ 0 on the interval [ a, ∞ ) .

Then the comparison theorem provides the following:

∞ ∞

a. If ∫ f(x) dx converges then so does I = ∫ g(x) dx

a a

∞ ∞

b. If ∫ g(x) dx diverges then so does I = ∫ f(x) dx

a a

Reasoning: If the area under the larger function, f(x), is finite, then the area under the

smaller function, g(x), must also be finite. (converges) Likewise, if the area under the

smaller function, g(x), is infinite, then the area under the larger function, f(x), must

also be infinite. (diverges).

Strategy: Examine f(x) and reason whether it might converge or diverge.

Then pick g(x) appropriately.

Note: The actual value of the improper integral, if it converges, is not determined.

Click here for examples.

Return to Notes for Calculus 2

Copyright © 2017 Richard C. Coddington

All rights reserved.