Barrons AP Calculus - David Bock

Neither limit exists; the integral diverges.

NOTE: This example demonstrates how careful one must be to notice a discontinuity at an interior

point. If it were overlooked, one might proceed as follows:

Since this integrand is positive except at zero, the result obtained is clearly meaningless. Figure N7–

26 shows the impossibility of this answer.

FIGURE N7–26

THE COMPARISON TEST
We can often determine whether an improper integral converges or diverges by comparing it to a
known integral on the same interval. This method is especially helpful when it is not easy to actually
evaluate the appropriate limit by finding an antiderivative for the integrand. There are two cases.

(1) Convergence. If on the interval of integration f (x) ≤ g(x) and is known to converge, then

also converges. For example, consider We know that converges. Since

the improper integral must also converge.

(2) Divergence. If on the interval of integration f (x) ≥ g(x) and is known to diverge, then

also diverges. For example, consider We know that diverges. Since sec x ≥

1, it follows that hence the improper integral must also diverge.

BC ONLY

EXAMPLE 30

Determine whether or not converges.

SOLUTION: Although there is no elementary function whose derivative is e−x^2 , we can still show

that the given improper integral converges. Note, first, that if x 1 then x^2 x, so that −x^2 −x and

e−x^2 e−x. Furthermore,