Improper integrals with infinite limits of integration: Continuous intervals

We call an improper integral a definite integral whose limits of integration are open intervals that go from minus infinity to infinity.

Definition of improper integrals with infinite limits of integration

[1] If f is continuous on the interval [a,∞), then:

[2] If f is continuous on the interval (-∞,b], then:

[3] If f is continuous on the interval (-∞,∞), then:

Where c is any real number.

In the first two cases, the integral converges if the limit exists, otherwise the improper integral diverges.

In the third case, the improper integral on the left diverges if any of the improper integrals on the right diverge.

Example of convergent or divergent improper integral with continuous intervals

Evaluate the following improper integral:

Since this integral is of the type:

then it is solved as follows:

As:

Then:

We apply the fundamental theorem of calculus and then solve the limit:

What does it mean that the result gave us infinity?

When we have in the integrand dx/x is because we are finding the area of f(x) = 1/x in the interval that goes from 1 to infinity, which means that the area under the curve is infinite, that is, its value is very large, we can corroborate this in the following graph generated with GeoGebra software:

In the interval from zero to infinity the function f(x) = 1/x has an infinite area.

Since the improper integral the limit exists and is infinite then the improper integral is convergent.

Bibliographic Reference

Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006