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. 

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Teaching Tips

Introducing the concept of u-substitution 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. 

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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. 

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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!

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Student Misconceptions

Two common errors appear with u-substitution 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.

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Secondly, emphasize that the inner function does not change when u-substitution 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(5x-7) still contains the expression (5x – 7) as an inner function.