In this lesson we are going to take a look at Improper Integration. The idea of improper integration is the function that we are trying to integrate has either a horizontal or a vertical asymptote. So, first, we are going to see how does it look like when we have horizontal or vertical asymptote. We will do some rough sketches for better comprehension. Also, we will see how to solve an asymptote's integral by cutting it off at a certain point. Then we will do some examples. Later on we will see that a few integrals come up very often, and because they are improper integrals, they should be remembered.
is often useful to graph the function youre integrating to get a
general idea of its shape and whether it is positive or negative.
you usually cant eyeball the difference between finite and
infinite area. You need to do the integral and take the limit to be
that infinite positive and infinite negative areas do not cancel.
When we have both, we just say that the integral diverges.
you can make a substitution and convert an integral into one of the
It is worth memorizing the chart for which values of p make
these integrals converge or diverge. These integrals will also be
useful for examining infinite series in later sections.
out for hidden ambushes when the function is discontinuous or
its denominator is zero in the middle of your interval. In these
cases, split the integral up around the discontinuity and evaluate
improper integrals on both sides.
you get something that is difficult to integrate, you may need to
use the Comparison Test with an easier integral (often
A common comparison is to use the fact that −1 ≤ sin x ≤
1, and similarly for cos x.
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.