Refresher - Convergent and Divergent Integrals
limt→∞∫taf(x) dx
exists and is equal to L if the integrals under the limit exist for all sufficiently large t, and the value of the limit is equal to L.
It is also possible for an improper integral to diverge to infinity. In that case, one may assign the value of ∞ (or -∞) to the integral. For instance
limb→∞∫b11x dx=∞.
However, other improper integrals may simply diverge in no particular direction, such as
limb→∞∫b1xsin x dx,
which does not exist, even as an extended real number.
A limitation of the technique of improper integration is that the limit must be taken with respect to one endpoint at a time. Thus, for instance, an improper integral of the form
∫∞-∞f(x) dx
is defined by taking two separate limits; to wit
∫∞-∞f(x) dx=lima→-∞limb→∞∫baf(x) dx,
provided the double limit is finite. By the properties of the integral, this can also be written as a pair of distinct improper integrals of the first kind:
lima→-∞∫caf(x) dx+limb→∞∫bcf(x) dx
where c is any convenient point at which to start the integration. It is sometimes possible to define improper integrals where both endpoints are infinite, such as the Gaussian integral ∫∞-∞e-x2 dx=√π . But one cannot even define other integrals of this kind unambiguously, such as ∫∞-∞x dx , since the double limit diverges: lima→-∞∫cax dx+limb→∞∫bcx dx