Integration by Parts

Active learning works for integration by parts because the strategic choice of u and dv requires repeated practice and immediate feedback. Students benefit from manipulating the formula directly rather than passively observing examples. The hands-on activities build confidence in decision-making, which is essential for solving complex integrals.

• JC 2
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Derive the integration by parts formula from the product rule of differentiation.
2. Strategically select 'u' and 'dv' from a product of functions to simplify integration.
3. Apply the integration by parts formula to calculate the integrals of various functions, including those involving logarithmic and inverse trigonometric functions.
4. Evaluate the effectiveness of integration by parts compared to other integration methods for specific problems.

Problem-Based Learning

⏱ 30 min·Pairs

Card Sort: u and dv Selection

Prepare cards with integrals, possible u choices, and dv options. In pairs, students match components, justify choices, then compute the integral. Discuss as a class which pairings work best and why.

Prepare & details

Explain the derivation of the integration by parts formula.

Facilitation Tip: During the Card Sort, circulate and ask groups to justify their u/dv choices aloud to uncover their reasoning.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 25 min·Small Groups

Relay Race: Step-by-Step Integration

Divide class into teams. Each student solves one step of an integration by parts problem on a board, passes to teammate. First team to correct answer wins; review strategies afterward.

Prepare & details

Analyze the strategic choice of 'u' and 'dv' in integration by parts.

Facilitation Tip: In the Relay Race, assign roles for each step to ensure every student participates in building the solution.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Problem-Based Learning

⏱ 35 min·Individual

Tabular Method Workshop

Provide integrals suited for tabular method. Individually, students create tables for differentiation and integration columns, then assemble results. Pairs swap and verify.

Prepare & details

Construct the integral of a product of two functions using integration by parts.

Facilitation Tip: For the Tabular Method Workshop, provide pre-filled examples to demonstrate the process before having students create their own.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Jigsaw

⏱ 40 min·Small Groups

Derivation Jigsaw

Cut product rule derivation into puzzle pieces. Small groups reassemble, derive integration by parts formula, and test on sample integrals. Present to class.

Prepare & details

Explain the derivation of the integration by parts formula.

Facilitation Tip: During the Derivation Jigsaw, have students compare their steps side-by-side to spot errors in the product rule conversion.

Setup: Flexible seating for regrouping

Materials: Expert group reading packets, Note-taking template, Summary graphic organizer

• Understand
• Analyze
• Evaluate
• Relationship Skills
• Self-Management

Teaching This Topic

Teach integration by parts by starting with simple examples like ∫x e^x dx, where the choice of u and dv is obvious. Emphasize that the goal is to reduce the integral to a simpler form, not just apply the formula mechanically. Research shows that students grasp the technique faster when they derive the formula themselves, so include a brief derivation activity. Avoid overwhelming them with complex integrals early; build from one-step to multi-step problems gradually.

What to Expect

By the end of these activities, students should confidently select u and dv, apply the formula correctly, and recognize when multiple applications are needed. They should also articulate why certain choices simplify or complicate the integral. Success includes the ability to explain their reasoning clearly to peers.

Watch Out for These Misconceptions

Common Misconception

During Card Sort: u and dv Selection, watch for students defaulting to polynomials as u without testing other options.

What to Teach Instead

Have students test at least two different u choices for each integral during the card sort and record which choices simplify the remaining integral. Guide them to notice that logarithmic or inverse trig functions often work better as u.

Common Misconception

During Relay Race: Step-by-Step Integration, watch for students assuming the formula applies only once.

What to Teach Instead

In the relay, include an integral like ∫x^2 e^x dx that requires two applications of integration by parts. Have students label each step clearly to reinforce that repeated applications are common.

Common Misconception

During Tabular Method Workshop, watch for students dismissing integrals that loop back on themselves.

What to Teach Instead

During the workshop, include an example like ∫e^x sin(x) dx and ask students to solve it step-by-step. Have them identify when the integral returns to the original form and discuss how to solve it algebraically.

Assessment Ideas

Quick Check

After Card Sort: u and dv Selection, present students with three integral problems: ∫x e^x dx, ∫ln(x) dx, and ∫sin(x) cos(x) dx. Ask them to identify which integral is best suited for integration by parts and to explain their choice of 'u' and 'dv' for one of the suitable integrals.

Discussion Prompt

After Relay Race: Step-by-Step Integration, pose the question: 'When might integration by parts lead to a more complicated integral than the original? Provide an example and explain why the choice of 'u' and 'dv' was strategic, even if it didn't immediately simplify the problem.' Facilitate a class discussion on common errors and effective strategies.

Exit Ticket

After Tabular Method Workshop, give students the integral ∫x^2 sin(x) dx. Ask them to write the formula for integration by parts, identify their chosen 'u' and 'dv', and write the resulting expression after one application of the formula.

Extensions & Scaffolding

• Challenge advanced students with integrals like ∫e^x sin(x) dx, where two applications of integration by parts loop back to the original integral, requiring algebraic manipulation to solve.
• For struggling students, provide partially completed solutions in the Tabular Method Workshop to focus on the process rather than the setup.
• For extra time, explore integrals like ∫ln(x) dx or ∫arctan(x) dx, which are classic examples where integration by parts is the only viable method.

Key Vocabulary

Integration by Parts	A technique for integrating products of functions, derived from the product rule for differentiation. It transforms an integral into a potentially simpler one.
Product Rule	The rule in differentiation stating that the derivative of a product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first.
u and dv	In the integration by parts formula, 'u' represents a function chosen to be differentiated, and 'dv' represents the remaining part of the integrand, chosen to be integrated.
du and v	After choosing 'u' and 'dv', 'du' is the differential of 'u', and 'v' is the integral of 'dv'.