Integration using Substitution (Topic 6.9)
Unit 6  Day 13 and 14
Unit 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Day 11
Day 12
Day 1314
Day 15
Day 16
Day 17
Day 18
Day 19
Day 20
Day 21
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Learning Objectives

Use usubstitution to find antiderivatives of composite functions
Success Criteria

I can determine when usubstitution is necessary for evaluating a definite integral

I can evaluate a definite integral using usub

I can rewrite the limits of integration when doing usubstitution
Quick Lesson Plan
Overview
A series of integrals opens this lesson on substitution. Most of the integrals can be solved using the basic rules of integration learned in Topic 6.8. However, students soon discover that some of the integrals cannot be resolved so easily: their tool for evaluating integrals has been broken and they need to devise a new rule! This experience serves as the motivation for the development by the students themselves of substitution techniques.
Teaching Tips
Introducing the concept of usubstitution is a more organic and meaningful lesson when students themselves realize the need for a new integration technique. Connect the notation from our earlier lesson on the Chain Rule to help students identify when the derivative of an inner function is necessary.
We spent two days on this EFFL lesson to allow students time to process the complex process of selecting an “inner” function for u and the associated derivative, du. Some instructors may be able to complete the entire lesson in one class period, but you should also feel free to revisit the Important Ideas and the CYU problems the next day. After completing the entire lesson, allow students time to work together on assigned problems. Group conversations are important and can reveal to you common sources of confusion or frustration.
Exam Insights
Substitution techniques were cleverly tested with questions #12 and #90 on the released 2012 Practice Test. Present these questions to students only after a day or two of practice with substitution!
Student Misconceptions
Two common errors appear with usubstitution problems. First, students tend to assume that every integral with a 1 or a dx term in the numerator of a fraction must resolve itself to a natural log antiderivative. To redirect this, give them plenty of experience finding arctan(x) as an antiderivative.
Secondly, emphasize that the inner function does not change when usubstitution is applied properly. This can be illustrated by first asking students for the derivative of cos(5x – 7). The derivative of this function still contains the expression (5x – 7): the inner function is unchanged. Then show students that the antiderivative of cos(5x7) still contains the expression (5x – 7) as an inner function.