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u-Substitution Steps

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Description

The u-Substitution Steps MicroSim breaks down the u-substitution process into five explicit steps, helping students internalize the systematic procedure for evaluating integrals using substitution. Each step is revealed sequentially with color-coded highlighting that distinguishes u (blue) from du (green).

The u-Substitution Method

For an integral of the form \(\int f(g(x)) \cdot g'(x) \, dx\):

\[\text{Let } u = g(x), \quad du = g'(x) \, dx\]
\[\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du\]

Or in words: "Find the inside function, call it u, replace everything, integrate, then substitute back."

How to Use

  1. Select an example integral: Click one of the six preset integrals at the bottom
  2. Step through the process: Click "Next Step" to reveal each step of the substitution
  3. Or show all at once: Click "Show All" to see the complete solution immediately
  4. Verify your answer: Click "Verify" to see the derivative check confirming the answer
  5. Reset and try another: Click "Reset" to start over with the same integral, or select a new one

The Five Steps

Step Action Example with \(\int 2x \cos(x^2) \, dx\)
1 Identify u (inside function) \(u = x^2\)
2 Calculate du \(du = 2x \, dx\)
3 Rewrite in terms of u \(\int \cos(u) \, du\)
4 Integrate in u \(\sin(u) + C\)
5 Back-substitute \(\sin(x^2) + C\)

Visual Features

  • Color coding: Blue highlights u throughout, green highlights du
  • Animated progression: Steps fade in smoothly as you advance
  • Current step highlighting: A pulsing border shows which step you are on
  • Verification panel: Shows the derivative check to confirm your answer
  • Animated arrow: Points to the next step to reveal

Delta Moment

"u-substitution is like putting on a disguise. The integral looks complicated, but once you substitute, it becomes something you already know how to handle. Then you unmask at the end. Very sneaky, very satisfying!"

Lesson Plan

Learning Objectives

By the end of this activity, students will be able to:

  1. Apply u-substitution systematically to evaluate integrals
  2. Identify the correct choice of u in an integrand
  3. Execute the five-step substitution procedure
  4. Implement u-substitution across various function types (trig, exponential, polynomial, radical)

Grade Level

High School (AP Calculus AB/BC) and Undergraduate Calculus I

Duration

15-20 minutes for initial exploration; can be revisited for practice

Prerequisites

Students should be familiar with:

  • Basic antiderivative rules (power rule, trig integrals, exponential integrals)
  • The chain rule for differentiation
  • The concept of composite functions
  • Differential notation (du, dx)

Activities

Activity 1: Recognizing the Pattern (5 minutes)

Work through the first two examples:

  1. \(\int 2x \cos(x^2) \, dx\) -- the derivative of the inside function appears directly
  2. \(\int (2x+3)^4 \, dx\) -- need to solve for dx in terms of du

For each, identify: What is u? What is du? Does du appear in the integral?

Activity 2: Building Complexity (10 minutes)

Move to the remaining examples and work through each:

  1. \(\int x\sqrt{x^2+1} \, dx\) -- only part of du appears (constant adjustment needed)
  2. \(\int \sin^3(x)\cos(x) \, dx\) -- recognizing trig substitution patterns
  3. \(\int e^{3x} \, dx\) -- exponential with linear inner function
  4. \(\int \cos(x)/\sin(x) \, dx\) -- rational form leading to logarithm

Compare Step 2 across examples: when does du appear exactly vs. needing adjustment?

Activity 3: Verification Practice (5 minutes)

For each example, click "Verify" and trace through the chain rule differentiation:

  1. Does differentiating the answer give back the original integrand?
  2. How does the chain rule connect to u-substitution?

Discussion Questions

  1. Why is u-substitution called the "reverse chain rule"? (Answer: It undoes the chain rule pattern in the integrand)
  2. What makes a good choice for u? (Answer: Look for a function whose derivative also appears in the integrand)
  3. What happens if you choose the wrong u? (Answer: The substitution won't simplify the integral; try a different choice)
  4. Why do we always verify by differentiating? (Answer: Integration is harder than differentiation, so checking catches errors)

Assessment

Quick Check: Without using the MicroSim, identify u and du for:

  1. \(\int 3x^2 \sin(x^3) \, dx\)
  2. \(\int \frac{e^x}{1 + e^x} \, dx\)
  3. \(\int x(x^2 - 4)^6 \, dx\)

Exit Ticket: Evaluate \(\int \frac{\cos(\ln x)}{x} \, dx\) showing all five steps of your work.

Common Mistakes to Address

Mistake Example Correction
Forgetting to substitute dx \(\int (2x+3)^4 \, dx = \int u^4\) Must replace dx with du/2: \(\frac{1}{2}\int u^4 \, du\)
Not back-substituting Final answer left as \(\frac{u^5}{10} + C\) Must replace u: \(\frac{(2x+3)^5}{10} + C\)
Variable adjustment error Putting x inside the integral after substitution After substitution, no x should remain
Wrong choice of u Choosing \(u = \cos(x^2)\) for \(\int 2x\cos(x^2) \, dx\) Choose the inside function: \(u = x^2\)

References

  1. u-Substitution - Khan Academy - Video explanations and practice problems

  2. Integration by Substitution - Paul's Online Math Notes - Detailed derivation and examples

  3. p5.js Reference - Documentation for the p5.js library used in this visualization