Integration by Substitution

Active learning works because integration by substitution requires students to think flexibly about composite functions and their derivatives. By moving between algebraic manipulation and conceptual reasoning, students solidify their understanding of the chain rule in reverse, which is essential for recognizing when substitution applies.

• JC 2
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Analyze the structure of composite functions to identify appropriate substitution candidates.
2. Explain the relationship between the chain rule and integration by substitution.
3. Construct the integral of a composite function by performing a valid u-substitution.
4. Calculate the definite integral of a composite function using substitution and adjusted limits.
5. Verify the result of an integration by substitution problem through differentiation.

Collaborative Problem-Solving

⏱ 30 min·Pairs

Pair Practice: u-Selection Challenge

Pairs receive cards with 10 composite integrals. They select u, write du, and outline substitution steps on mini-whiteboards. Switch cards with another pair after 5 minutes for peer review and revision. Conclude with class share-out of trickiest examples.

Prepare & details

Analyze when integration by substitution is an appropriate technique.

Facilitation Tip: During the Pair Practice: u-Selection Challenge, circulate and ask each pair to explain their choice of u aloud, reinforcing the chain rule connection.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Small Group Relay: Substitution Chain

Divide class into groups of 4. Each group lines up; first student solves first step of integral (choose u), tags next for du and substitution, and so on until back-substitution. First group to finish correctly wins. Repeat with 3-4 integrals.

Prepare & details

Explain the chain rule in reverse as the basis for substitution.

Facilitation Tip: In the Small Group Relay: Substitution Chain, provide each group with a dry-erase board to write each step clearly, making errors visible and easier to address.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 25 min·Whole Class

Whole Class: Integral Matching Game

Project 12 integrals and 12 substitution setups. Students stand and point to matches as you reveal hints. Discuss mismatches as a class, then have volunteers solve one fully on board. Use for definite integrals next round.

Prepare & details

Construct the integral of a composite function using substitution.

Facilitation Tip: For the Whole Class: Integral Matching Game, assign each student a role, such as ‘u-identifier’ or ‘limit adjuster,’ to ensure everyone participates actively.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 20 min·Individual

Individual: Guided Substitution Worksheet

Provide worksheets with partially worked integrals. Students fill gaps: choose u, compute du, integrate, back-substitute. Include 8 problems escalating in complexity, with space for self-check by differentiating answers.

Prepare & details

Analyze when integration by substitution is an appropriate technique.

Facilitation Tip: With the Individual: Guided Substitution Worksheet, include a ‘think-aloud’ section where students write their reasoning for each step before performing calculations.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Experienced teachers approach this topic by first grounding substitution in the chain rule, using visuals like function machines to show composition and decomposition. Avoid rushing to procedures—instead, emphasize the ‘why’ through concrete examples. Research suggests that students benefit from multiple representations, so pair algebraic steps with graphical or numerical checks to deepen understanding.

What to Expect

Successful learning looks like students confidently identifying the inner function for substitution, correctly computing du, and accurately back-substituting to the original variable. They should also recognize when substitution is appropriate and when other methods may be better.

Watch Out for These Misconceptions

Common Misconception

During Pair Practice: u-Selection Challenge, watch for students who omit the derivative factor after substitution.

What to Teach Instead

Prompt partners to verbalize the chain rule link by asking, ‘What is the derivative of your chosen u? How does it relate to the integrand?’ This forces them to include du/dx explicitly.

Common Misconception

During Small Group Relay: Substitution Chain, watch for students who select u as the entire integrand instead of the inner function.

What to Teach Instead

Have groups compare their u choices on the board and discuss which one simplifies the integral most effectively. Peer teaching during the relay helps clarify nested function identification.

Common Misconception

During Whole Class: Integral Matching Game, watch for students who do not adjust limits for definite integrals.

What to Teach Instead

Use the matching game to highlight this error visually—assign one student to act as the ‘limit checker’ who must adjust limits before proceeding. This collaborative role reinforces the importance of limit changes.

Assessment Ideas

Quick Check

After Whole Class: Integral Matching Game, present students with three integrals: ∫ x(x^2 + 1)^3 dx, ∫ sin(x)cos(x) dx, and ∫ e^x dx. Ask them to identify which integral(s) can be solved using substitution and to explain their reasoning for each.

Exit Ticket

After Individual: Guided Substitution Worksheet, provide students with the integral ∫ 2x * sqrt(x^2 + 5) dx and ask them to: 1. Identify the substitution u, 2. Calculate du, 3. Write the integral in terms of u, 4. State the final answer after back-substitution.

Discussion Prompt

During Small Group Relay: Substitution Chain, pose the question: ‘When might integration by substitution NOT be the most efficient method for solving an integral?’ Have students discuss in pairs and share scenarios where other integration techniques might be preferable.

Extensions & Scaffolding

• Challenge students to create their own integral pair where substitution is the most efficient method, then trade with a partner to solve.
• For students who struggle, provide a bank of integrals with partially completed u and du calculations to scaffold the process.
• Deeper exploration: Have students investigate integrals where substitution leads to circular reasoning, such as ∫ sin(x)cos(x) dx, and discuss why this happens.

Key Vocabulary

u-substitution	A technique for simplifying integrals by replacing a part of the integrand with a new variable, u, and its differential, du.
composite function	A function that is formed by applying one function to the result of another function, often written as f(g(x)).
chain rule	A rule in calculus for differentiating composite functions, stating that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x).
differential	An infinitesimally small change in a variable, denoted as dx or du, used in integration and differentiation.