Techniques of Integration: Substitution

Active learning works for u-substitution because students often struggle with recognizing composite functions and translating them into the correct substitution. Hands-on activities let them practice identifying inner functions, computing du, and rewriting integrals, which builds the procedural fluency needed to succeed with integration techniques.

• Year 12
• Mathematics
• 4 activities
• 20 min–40 min

Learning Objectives

1. Analyze the relationship between the chain rule for differentiation and the u-substitution method for integration.
2. Evaluate indefinite integrals using the u-substitution technique, including selecting an appropriate substitution.
3. Calculate definite integrals using u-substitution, applying either back-substitution or limit substitution.
4. Design a novel integral problem that necessitates u-substitution and justify the choice of the substitution variable.
5. Compare and contrast the steps for solving indefinite and definite integrals using u-substitution.

Peer Teaching

⏱ 30 min·Pairs

Pairs Relay: Substitution Chain

Pair students and provide a starting integral on the board. One student writes the u-substitution and du, passes to partner for rewriting the integral, then back for integration and back-substitution. Time each relay for 2 minutes, then discuss solutions as a class.

Prepare & details

Analyze how the chain rule for differentiation relates to the u-substitution method for integration.

Facilitation Tip: During Pairs Relay, have students alternate roles every two steps to keep both partners engaged and accountable for accurate substitution.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Peer Teaching

⏱ 25 min·Small Groups

Small Groups: Card Sort Match-Up

Prepare cards with integrals, possible u values, du expressions, and antiderivatives. Groups sort and match sets, then verify by differentiating results. Extend by creating mismatched sets for peers to fix.

Prepare & details

Justify the steps involved in performing a u-substitution to evaluate both indefinite and definite integrals.

Facilitation Tip: For Card Sort Match-Up, provide a mix of simple and complex integrals to push students to think beyond obvious substitutions.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Peer Teaching

⏱ 40 min·Small Groups

Whole Class: Gallery Walk Critique

Assign each group an integral to solve using u-substitution on poster paper, including justifications. Groups rotate to critique others' work with sticky notes, focusing on choice of u and limit changes for definite integrals.

Prepare & details

Design an integral that requires u-substitution and explain the reasoning behind the choice of substitution variable.

Facilitation Tip: In Gallery Walk Critique, assign specific feedback prompts (e.g., 'Explain the substitution step clearly') to guide constructive comments among groups.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Peer Teaching

⏱ 20 min·Individual

Individual: Design Challenge

Students create three integrals requiring specific u-substitutions, swap with a partner to solve, then explain their reasoning in a short write-up. Collect for class sharing of clever examples.

Prepare & details

Analyze how the chain rule for differentiation relates to the u-substitution method for integration.

Facilitation Tip: During Design Challenge, require students to include a written explanation of their chosen u and du alongside their integral rewrite for metacognitive reflection.

Setup: Presentation area at front, or multiple teaching stations

Materials: Topic assignment cards, Lesson planning template, Peer feedback form, Visual aid supplies

• Understand
• Apply
• Analyze
• Create
• Self-Management
• Relationship Skills

Teaching This Topic

Teachers should emphasize the connection between u-substitution and the chain rule by asking students to verbalize how du relates to the original derivative. Avoid rushing through examples—instead, model pauses to check each step aloud. Research suggests that students benefit from seeing both successful and incorrect substitutions side by side to deepen understanding of why certain choices work better.

What to Expect

By the end of these activities, students should confidently choose u, compute du correctly, and rewrite both indefinite and definite integrals without errors in variable or limits. They should also explain their choices and justify substitutions during peer discussions.

Watch Out for These Misconceptions

Common Misconception

During Pairs Relay, watch for students who skip rewriting dx in terms of du or fail to adjust the integral accordingly.

What to Teach Instead

Require partners to pause after each substitution step and ask, 'Does this rewrite match the original integral in terms of u and du?' If not, they must revise together before moving forward.

Common Misconception

During Card Sort Match-Up, watch for students who ignore the limits of integration when matching substituted integrals to their original forms.

What to Teach Instead

Have groups explain their matching choices aloud, focusing on why the new limits correspond to the original variable’s range. Peer questioning helps catch mismatches quickly.

Common Misconception

During Pairs Relay, watch for students who choose u as the entire integrand instead of the inner function.

What to Teach Instead

Prompt partners to ask, 'Is du simple and linear?' If not, they must reconsider their u choice. Discuss as a class why inner functions (like 3x in sin(3x)) simplify the process.

Assessment Ideas

Quick Check

After Pairs Relay, distribute an exit ticket with ∫ 2x cos(x^2) dx and ask students to identify u, compute du, and rewrite the integral. Collect responses to assess procedural accuracy before moving to the next activity.

Discussion Prompt

During Gallery Walk Critique, have students post their solutions to a definite integral problem (e.g., ∫ from 0 to π/2 of cos(2x) dx) and rotate to provide feedback on whether limits were changed or substituted back correctly.

Exit Ticket

After Design Challenge, ask students to submit their integral rewrite for ∫ (3x^2 + 1) / (x^3 + x) dx along with a sentence explaining their choice of u. Use this to assess both procedural skill and conceptual justification.

Extensions & Scaffolding

• Challenge students who finish early to create a composite function integral of their own, then trade with a partner to solve it.
• For students who struggle, provide partially completed substitution steps (e.g., 'u = ___, du = ___ dx') to scaffold the process.
• Deeper exploration: Ask students to derive the general formula for integrating functions of the form ∫ f(ax + b) dx using u-substitution, then verify with examples.

Key Vocabulary

u-substitution	An integration technique that simplifies an integral by replacing a part of the integrand with a new variable, typically denoted by 'u'. It is the reverse of the chain rule.
differential (du)	The differential of the substitution variable 'u', calculated by differentiating 'u' with respect to the original variable (e.g., x) and multiplying by dx. For example, if u = f(x), then du = f'(x) dx.
composite function	A function that is formed by applying one function to the result of another function. U-substitution is particularly useful for integrating composite functions.
integrand	The function that is being integrated. In u-substitution, the integrand is rewritten in terms of 'u' and 'du'.