Definition. An improper integral is an integral where one or the two limits of integration are infinite. An integral is also improper when the function has an infinite discontinuity. Let's look at the two scenarios:
I. One or two of the limits of integration are infinite.
a. One of the limit of integration is finite,
If f is continuous over the interval [a, +∞[, then
The integral represents the area between the verticals passing by a and l. As l approaches infinity, the area under the curve spreads to infinity. It's fair to say that this situation represents .
We can define as the limit of when l approaches infinity.
We can then set a rule to calculate the improper integral where the lower limit is finite and the upper limit is infinite. In this case we have to replace the infinite limit by l and find the limit of when l approaches infinity.
Example 1. Calculate
Let's replace infinity by l and calculate the limit of the finite integral when l approaches infinity.
Let's say that we have to calculate an integral over an interval ]a, b[ where the function f is continued. If the function isn't continued at a point c of the interval, we evaluate the integral over the intervals ]a,c[∪]c,d[.
The function is discontinued at x = 1. Therefore we integrate it over the 2 intervals ]0,1[∪ ]1.4[.
Let's calculate the first integral:
= ln │1⁻-1│-ln1
= ln o⁺
Evaluate. Tell if the integral converges or diverges:
Center for Integral Development