Authentic Lessons for 21st Century Learning

I Sub, U-Sub, We All Sub Again

Integration: U-Subsitution, Part 2

Cacey Wells | Published: August 4th, 2022 by K20 Center

  • Grade Level Grade Level 12th
  • Subject Subject Mathematics
  • Course Course AP Calculus
  • Time Frame Time Frame 75 minutes
  • Duration More 2-3 class periods


This lesson is a follow-up to I Sub, U-Sub, We All Sub. In the previous lesson, students worked with the teacher to create an anchor chart of what "worked" and what "didn't work" for anti-differentiating derivatives of functions. The goal of this lesson is to build on that knowledge in order for students to begin seeing that what "worked" worked for a specific reason. Coupling this notion with the connection to what "didn't work" will allow students to see how U-substitution is a manipulated form of what did work in Part 1.

Essential Question(s)

How can we undo the chain rule? In other words, how can we find the antiderivative of a function that has been differentiated using the chain rule?



Using the anchor chart from the I Sub, U-Sub, We All Sub lesson, teachers and students identify what worked and what didn't work. Students label part of the equation as u and the other part as du.


Students explore the idea that part of the derivative of the chain rule looks similar to the derivative of another part of the function. Students then explore how to identify these parts as u and du.


Clarify what u represents and what du represents. Students then articulate that the integral of u du, looks like the integral of x dx.


Practice with these new types of problems in a scavenger hunt, identifying u and du, and then finding the antiderivative.


What? So What? Now What?


  • Paper and pencil


Display the anchor chart from the I Sub, U-Sub, We All Sub lesson.

Have students articulate which parts of the equation look like the derivative of the other part (u and u').

Pose the question, "What if we replace u' with du? Would that change anything?"

On the anchor chart, replace u' with du in each of the equations that "didn't work."


Distribute the Exploring U-Sub handout to each student. Ask students to work in pairs to go through the problems.

As students progress through the activity, check in with each pair to help clarify questions that they might have.


After students have finished both parts of the Explore activity, move on to the Explain activity, which is intended to clarify misconceptions and make sure students all understand the concepts.

Project or print the questions in the Explain Handout and ask students to explain each question. Give students ample time to work with their partner to explain their reasoning.


Have students complete the scavenger hunt, either individually or in pairs. Be sure to make yourself available if students have questions.


Display the attached Evaluate document to have students reflect on their learning and try a challenge problem.

Give students two or three minutes each to complete the first two questions. Allow students to explore the final question and use it as a starting point for conversations about manipulating the substitution.

And...we're done!