Identify integrands that emerge from product rule derivatives.
Identify the appropriate expressions for f and g' and recognize how the choices for f and g' simplify the resulting integrand.
Apply integration by parts to evaluate definite and indefinite integrals.
Quick Lesson Plan
In this lesson, students learn a new integration technique to add to their toolbox: integration by parts. The activity starts with having students recall the product rule, as well as the inverse relationship between differentiation and integration. Questions 3 and 4 lead students through some algebraic manipulation that will allow them to write an expression for the integral of x cos x which is a function that they couldn’t previously integrate!
Have students pause after question 4 for a whole-class debrief. Today’s margin notes are color-coded so students can see how each expression results from the original product rule equation in question 1. The biggest thing for students to see is that the expression x cos x represents a part of the product rule derivative. The question for students is ‘which function would have x cos x as one term of its derivative?’ This will lead them to think about a product where the first function is x and the second function has a derivative of cos x since the first part of the product rule is given by fg’. Of course, it may be written in reverse order where the first factor is the derivative and the second factor is the original function, but it is unnecessary to point this out to students at this point.
After the margin notes for questions 1-4, have students reconvene in their groups to try questions 5 and 6. These questions are all about having students recognize integrals that can be solved using this same method. In other words, they are looking for integrand functions that look like one function times the derivative of another. There are a few tricky ones in there, so expect to hear some lively discussion among groups!
When debriefing questions 5-6, focus on the difference between integrals that are solved with u-sub and integrals that are solved with integration by parts. The big take-away is that integrals that can be solved with u-sub represent the derivative of a composite function whereas integrals that can be solved with integration by parts represent part of the derivative of a product of functions.
How are questions 1 and 2 related?
Are there any other strategies you can think of to integrate x cos x?
(Question 4) What seems to be the strategy here?
What do you notice about the other integral expression (integral of sin x)?
How did re-writing the integral make it easier to solve?
How can you recognize integrals that can be solved with this strategy?
What made you choose u-sub for this one?
Expect that it will take students time to confidently use integration by parts. Avoid the tendency to over-proceduralize the steps. Identifying f and g’ can be tricky, but the worst that can happen is that they choose wrong and they realize their integral has not become any easier to integrate! This is the way we encourage students to learn integration by parts. Over time, they will start to build intuition behind which function is f and which function should be g’, knowing that they will have to find the derivative of f and the antiderivative of g’ and that the product of these will have to be integrated.
The QuickNotes also introduce integration by parts with a definite integral. The process is the same except that students must evaluate the antiderivative at the upper and lower limits of integration using the FTC.
Note that we choose to use f and g’ in this lesson instead of u and v’ as is used in many textbooks. We find that this helps students make better connections to the way they learned the product rule in Lesson 2.8.
As always, communication is key when responding to an integration by parts question on the AP exam. When students clearly label their “parts”, whether they use u/dv notation or f/g’ notation, the reader can easily follow the student’s thoughts. However, when the first step shown is the actual integration, the reader is left to decipher what the missing steps may have been. A good question that students should ask themselves is this: if I were scoring this work, would I be able to follow the entire process that was taken to solve this problem?