Handout Daily Prep 5.3 - Integration by Substitution

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Section Overview

Section Basic learning objectives

• Recognize the notation for an indefinite integral and state its meaning.

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• Determine the general antiderivative given a composite function whose “inner” function is linear.

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Section To prepare for class

• Read motivating questions and the introduction to section 5.3 (up until Preview Activity 5.3.1).

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• 📝 [Submit] Do Preview Activity 5.3.1 parts (a) and (b) numbers (i) and (ii).

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  • (Optional) Watch video solution to Preview Activity 5.3.1 (7:23).

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• Read section 5.3.2 up to Activity 5.3.2.

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• Prompt Copilot “Walk me through an example in which \(u\)-substitution is used to calculate an indefinite integral.”

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• Read section 5.3.3.

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• Watch video Integration by Substitution Example 1 (3:43).

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• Watch video Integration by Substitution Example 2 (3:50).

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• Watch video Integration by Substitution Example 3 (4:42).

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• Watch video Integration by Substitution Example 4 (4:19).

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• Watch video Integration by Substitution Example 5 (3:40).

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• Do these problems.

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  1. 📝 [Submit] Explore the applet U-Substitition by Tim Brzezinski. It illustrates the idea of \(u\)-substitition. No matter the value of \(a\text{,}\) \(b\text{,}\) of how you position the slider, how are the areas of the green and blue regions related?

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  2. 1. Using the \(u\)-substitution \(u(x) = x^{3}\text{,}\) find \(\displaystyle \int 3x^{2} \cos(x^{3}) \ dx\text{.}\)

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     2. Repeat (a) using the applet Integration by Substitution by Ravinder Kumar. Note that you will need to enter the integrand and the function \(u(x)\text{.}\) It then gives step by step instructions. Note also that the applet uses \(v(x)\) rather than \(u(x)\text{.}\) Does your answer match that you found in (a)?

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     3. Use the applet to check the correctness of Example 5.3.2 in the text.

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• Watch video Quick Recap - Integration by Substitution (3:27).

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• 📝 [Submit] Do write a paragraph or two with your reflections on the value of each of the following as they appeared in the Daily Prep assignments this semester. These comments will be used to improve the Daily Prep assignments for the next semester.

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  1. The readings from the textbook

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  2. The interactive applets (often using GeoGebra)

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  3. The videos

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  4. The problems in the textbook (interactive and written)

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  5. The problems not in the textbook

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Section After class

• Read section 5.3.4.

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• Explore the applet Integration by Substitution by John Golden. Answer the single question of ‘Why are these definite integrals equal?’ In fact, write down both integrands and both limits of integration and then approximate each definite integral using a Riemann Sum via the applet Riemann Sums by J Mulholland. This should convince you that both definite integrals do indeed have the same value.

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• Watch video Integration by Substitution Example 6 (3:08).

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• Watch video Definite Integrals using Substitution (4:31).

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• Prompt Copilot “If the function \(u\) is not one-to-one, the method of \(u\)-substitution as applied to a definite integral may fail. Give me an example showing this.”

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• Do exercises 1-3 in section 5.3.6.

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• Do these problems.

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  1. Find \(\displaystyle \int x^{3} \sqrt{x^{4}+5}\ dx\) by letting \(u(x) = x^{4} + 5\text{.}\)

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  2. Find \(\displaystyle \int x\sqrt{x+2}\ dx\text{.}\)

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  3. Find \(\displaystyle \int \tan(x) \ dx\text{.}\)

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  4. Find \(\displaystyle \int \sec(x) \ dx\text{.}\) Hint: Multiply the integrand by \(\displaystyle \frac{\sec(x) + \tan(x)}{\sec(x) + \tan(x)}\text{.}\)

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• Read section 5.3.5 - summary.

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• Do exercises 7-10 in section 5.3.6.

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• Start working on the MOMwork (MyOpenMath) assignment for this section.

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Section Advanced learning objectives

• Use proper notation when using Integration by Substitution. In particular, fully convert an integral from \(x\)’s to \(u\)’s and back again, without mixing the two variables.

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• Use Integration by Substitution in unusual cases, such as those where there is not an obvious substitution.

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Section Additional suggestions

• Watch video U-Substitution for Antiderivatives (8:13).

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• Finish (if needed) Activity 5.3.2.

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  • Watch video solution to (a)-(c) of Activity 5.3.2 (4:08) and solution to (d)-(f) of Activity 5.3.2 (4:57).

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• Finish (if needed) Activity 5.3.3.

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  • (Optional) Watch video solution to Activity 5.3.3 (4:36).

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• Do more practice via the applet Integration by Substitution by R. Kumar. Clicking ‘Practice’ generates a new problem. Clicking ‘Show Answer’ can be used to check your work.

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Section Answers