Overview
This week, we see the last integration techniques we will cover in this course, starting with the method of partial fractions: This technique enables us to antiderive certain ratios of polynomials (i.e. rational functions), for which other methods (like substitution or integration by parts) might not apply. The idea is to replace the rational function by a polymomial plus the sum of simpler rational functions with denominators of at most degree 2.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.

Be able to do polynomial division in order to write a rational function as a polynomial plus a ratio of polynomials for which the degree of the numerator is less than that of the denominator.

Decompose a rational function into its partial fractions when the factors of the denominator are linear.

Use a partial fraction decomposition to integrate a rational function when the factors of the denominator are linear.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Decompose a rational function into its partial fractions when some of the factors of the denominator are irreducible quadratics.

Use a partial fraction decomposition to integrate a rational function when some of the factors of the denominator are irreducible quadratics.

Perform rationalizing substitutions when appropriate and then use the method of partial fractions to evaluate the integral.
To prepare for class

Watch at least one of the following two videos which review long division of polynomials: Since the notation for this algorithm is slightly different in French and English, you may want to rather watch the second video if you went to a French high school. Both notations are equally acceptable, just pick one and stick to it.

Do the first problem on the Preview Activity for this section (on WeBWorK if required by your teacher).

Watch the following video which explains how Partial Fraction Decomposition is done when the factors of the denominator are linear. Watch it until 10:10. If you are curious, the remaining part of the video shows how the method of partial fractions is applied when the denominator has irreducible quadratic factors (which you should look at after class, together with the video I've listed further below):

Watch the following video which shows how Partial Fraction Decomposition is used to integrate a ratio of polynomials:

Do Problems 2 and 3 on the the Preview Activity for this section (on WeBWorK if required by your teacher). Problem 2 has distinct linear factors, and Problem 3 has a repeated linear factor (if you are unsure how to decompose the rational expression into partial fractions, watch the previous video again).
After class

Watch the following video which shows how to integrate a rational function which has an irreducible quadratic in the factors of the denominator.

Watch the following video which shows how a rationalizing substitution can lead to a rational expression which can be integrated using the method of partial fractions: