Use u-substitution to find antiderivatives of composite functions
I can determine when u-substitution is necessary for evaluating a definite integral
I can evaluate a definite integral using u-sub
I can re-write the limits of integration when doing u-substitution
Quick Lesson Plan
Today’s Scavenger Hunt provides engaging practice with u-substitution. Students work in groups to solve eleven integrals, using the answer of each problem to locate the next station.
Prior to the Scavenger Hunt, as a warm-up activity, our students were presented with a definite integral which required substitution techniques. We allowed students to evaluate their integral and then encouraged them to confirm their result with NINT (Math:9) on their calculators. Those students who used integrands written in terms of u --- but evaluated with limits in terms of x --- were surprised to discover a discrepancy in their results. It is critical that students understand the source of the discrepancy.
The multiple-choice options on the Scavenger Hunt include answers obtained by common mistakes and many incorrect answers will be attractive to students. Teachers should move among the groups during this activity to ask guiding questions and provide support for struggling students.
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!
Students forget to reconcile the independent variable of the integrand with their limits of integration: a student who uses the original integrand should use the original limits of integration. However, if the integrand is rewritten in terms of u, the limits of integration must be values of the function u, as well. It would be valuable to occasionally require that students rewrite the original integral in terms of u. Remind them to find new limits of integration, too.