Which integral is improper

Otherwise, we say the improper integral diverges , which we capture in the following definition. First we compute the indefinite integral.

Definition 2. Otherwise, we say the improper integral diverges. Evaluate it if it is convergent. With Example 2. For good measure, here is one more example. We use the Comparison Test to show that it converges.

An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.

Some such integrals can sometimes be computed by replacing infinite limits with finite values. If decreases at least as fast as , then let.

If diverges as for , let. Jeffreys, H. Cambridge, England: Cambridge University Press, pp. Press, W. Weisstein, Eric W. An integral is also considered improper if the integrand is discontinuous on the interval of integration, which means that the function we're integrating has a discontinuity in the interval.

The discontinuity can also occur at either of the endpoints of the interval, or at both endpoints, and the function would still be considered to have a discontinuity in the interval, and would therefore still be an improper integral. In this video we'll discuss exactly what it takes to make an integral improper. What makes an integral improper?

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