New Week 4 – 7.4 Indefinite Integrals

Step 1: Week 4 kickoff meeting – Monday 10:10 am

(or watch the recording later)
• Agenda: Logistics, questions, lecture/discussion introducing the indefinite integral, families of anti-derivatives, and using the differentiation rules “backwards”

Step 2: Get started on the WebAssign homework

• Log into WebAssign from Canvas and start the Section 7.4 homework
• Be sure to use the e-book when you need help and to look at relevant definitions, theorems, and examples
• All of the problems on this assignment ask you to use integration formulas to solve integrals.
• KEY HINT: Some of the integrands may need to be simplified or rewritten using algbra before the integration formulas will apply.
• KEY HINT: Integration (anti-differentiation) is harder than differentiation; there isn’t always an obvious path or formula to apply. You may have to guess-and-check or try a number of different strategies.
• KEY HINT: You can always check your answer by differentiating it; you should get back the original integrand.
• Problems 1-6 are about simple algebraic functions and the “reverse power rule”; be sure to multiply out or simplify first if needed.
• Theorem 7.16 has the key integration formulas that you will need for these problems. See also Example 2.
• Watch *all* of the videos in this Khan Academy playlist on the Reverse Power Rule, including the Practice modules that are in that playlist. This playlist really closely follows exactly the skills you will need for the first part of your WebAssign homework.
• Note that according to Definition 7.15, indefinite integrals are families of anti-derivatives that differ by constants; thus you need to include “+C” in your answers here and for all integration problems in this section. See also Example 1.
• For some of these problems you’ll also implicitly be using Theorem 7.20; be sure you understand why and how you are using it, since a more abstract question about this aspect of these problems may appear on the quiz. See also Example 3.
• Problems 7-12 expand into the more complicated exponential, trigonometric, and inverse trigonometric functions that we have studied. Think about what you know about the derivatives of such functions — you’re basically doing it backwards here.
• See Theorems 7.17, 7.18, and 7.19 for the integration fomulas needed for these problems.
• See Examples 5(a) and 5(c) to see how to handle the extra constants in the integrands using a guess-and-check strategy and some clever algebra when needed.
• Watch *all* of the videos in this Khan Academy playlist on Common Indefinite Integrals, including the Practice modules in the playlist, especially if you feel that you need to brush up on your basic differentiation and anti-differentiation skills.
• Don’t forget that you can simplify algebraically before attempting to anti-differentiate, and that sometimes you may need to guess-and-check multiple times to solve the problem. See the Taalman/Kohn Calc Clip video for Section 7.4 in the Resources tab of your WebAssign dashboard, which is very similar to Problem 10 in your assignment.
• If you’re stuck on Problem 12 then you may find it helpful to watch this Turksvids video on Integrals that Result in Arctangent.
• Problems 13-15 require a little more thinking, and in particular need you to consider how to do the chain rule “in reverse”.  For these problems it is especially helpful to check your answers afterwards by differentiating.
• See Theorem 7.21, especially the formula in part (c) of that theorem. Make sure you understand how this formula is just a “backwards” interpretation of the chain rule.
• See Example 4, especially part (b) which uses the technique of recognizing the integrand as the result of a chain rule calculation. Notice that the answer is checked at the end by differentiating. See also Example 5(b).
• For a basic video example watch this Khan Academy video on the Reverse Chain Rule – but don’t worry about the end part when he uses the “u” notation or talks about “u-substitution” (that’s Section 8.1, which we will not be covering this term). Then watch the slightly harder example worked out in the Khan Academy video on a Reverse Chain Rule Example.
• Integration is tricky, and every problem is a little bit different. To give you an overview of a few strategies and help you prepare for this week’s quiz I made you this video: Similar integrals can require very different solving strategies.

Step 3: Take a break

• Take a nap. You deserve to take a nap and also it helps keep your body healthy and strong. If you’re tired then you need to rest. Encourage others in your family to get the rest that they need, even if they are impossibly busy (or impossibly bored, as the case may be).

Step 4: Get help if you need it… and finish the homework

• Attend the Zoom drop-in 4-6pm Thursday (link in Canvas)
• Or, if you can’t attend, watch the recording (link to be added in Canvas later)
• Agenda: Discuss wekly updates, get math help, ask Dr. Taalman anything
• If you need help now or at any point: For math questions, use Canvas Chat or Discussion, or contact the SMLC. For other questions text Dr. Taalman directly.

Step 5: Take the Mastery Quiz for Section 7.4

• When you’re ready, take the 7.4 Mastery Quiz in Canvas (and retake if needed)
• I recommend you take your first quiz attempt by Friday or Saturday