Educator

Improper Integration

Main formula:

Horizontal asymptote:

Vertical asymptote:

Hints and tips:

• It is often useful to graph the function you’re integrating to get a general idea of its shape and whether it is positive or negative.

• However, you usually can’t eyeball the difference between finite and infinite area. You need to do the integral and take the limit to be sure.

• Remember that infinite positive and infinite negative areas do not cancel. When we have both, we just say that the integral diverges.

• Often, you can make a substitution and convert an integral into one of the form or . It is worth memorizing the chart for which values of p make these integrals converge or diverge. These integrals will also be useful for examining infinite series in later sections.

• Watch out for “hidden ambushes” when the function is discontinuous or its denominator is zero in the middle of your interval. In these cases, split the integral up around the discontinuity and evaluate improper integrals on both sides.

• If you get something that is difficult to integrate, you may need to use the Comparison Test with an easier integral (often or ). A common comparison is to use the fact that −1 ≤ sin x ≤ 1, and similarly for cos x.