Lesson 9: 6.9 Integrating Using Substitution

How do we identify which part of an integrand should be defined as u?
What is the role of the differential du in the substitution process?
How does the "inside" function of a composite derivative relate to the "outside" function during integration?
When evaluating a definite integral with u-substitution, how and why must the limits of integration be adjusted?

u-Substitution
Change of Variables
Composite Function
Inner Function
Differential
Change of Limits

College Board AP Calculus CED: FUN-6.D (For composite functions, use substitution to find antiderivatives).

Mathematical Practices: MP 1 (Implementing Mathematical Processes), MP 2 (Connecting Representations).

Students learn to recognize integrands that are the result of a Chain Rule derivative. Students practice choosing a u, calculating du, and rewriting the entire integral in terms of u. The lesson places a heavy emphasis on "balancing constants" (multiplying and dividing by a scalar) and the crucial step of changing the $-boundaries to u-boundaries for definite integrals.

The purpose of 6.9 is to expand the student's "integration toolkit" beyond basic rules. U-substitution is the "Chain Rule in reverse." Mastery of this technique is essential because it appears in nearly every unit that follows, including differential equations, area between curves, and volumes of solids. It also teaches students to look for structural patterns in mathematics.

DOK 1 (Recall): Identifying the "inner function" u in a basic composition.
DOK 2 (Skill/Concept): Successfully performing u-substitution on an indefinite integral where the derivative du is present or differs only by a constant.
DOK 3 (Strategic Thinking): Evaluating a definite integral by transforming the limits of integration and justifying why the numerical result remains the same in both the x and u "worlds."

Support (Scaffolding):The "U-Du" T-Chart: Provide a workspace where students must explicitly write u = and du = in a side column before attempting to rewrite the integral
.Pattern Recognition Drill: Give a list of 10 integrals and have students only identify the u and du without actually solving the integral.
Limit Transformation Box: For definite integrals, provide a dedicated "Change of Limits" box on their worksheets to prevent the common error of using x-limits with u-integrands.

Extension (Enrichment):"Double Substitution" (Back-Substitution): Challenge students with problems where they must solve for x in terms of u to completely substitute the integrand.

Ap College Board Assessments