Integration by Substitution: Study Guide

Learning Objectives

• Identify the appropriate "inner function" $u$ to simplify complex integrands.
• Apply algebraic manipulation (constant alteration) to match the differential $du$.
• Solve integrals requiring substitution by eliminating original variables (solving for $x$).
• Evaluate definite integrals by accurately mapping the limits of integration from $x$-space to $u$-space.

Key Terms & Glossary

• $u$-Substitution: A mathematical technique used to evaluate integrals by reversing the chain rule.
• Integrand: The mathematical expression or function that is being integrated.
• Definite Integral: An integral with a defined start and end point (limits of integration), computing net area.
• **Differential ($du)**: An infinitesimally small change in the variable u$, crucial for variable transformation.

The "Big Idea"

[!IMPORTANT] $u$-Substitution is the Chain Rule in reverse. When you take the derivative of a composite function, you multiply by the derivative of the inside function. $u$-substitution seeks out that "inside function" and its derivative to un-do the process, transforming an impossible integral into a fundamental one.

Formula / Concept Box

Hierarchical Outline

• 1. Basics of Substitution
  • Identify $u$: Usually the quantity under a root, raised to a power, or in a denominator.
  • Find $du$: Differentiate $u$ with respect to $x$.
• 2. Substitution with Alteration
  • Scaling constants: Multiply or divide $du$ to match the $x$ terms exactly.
  • Pulling constants out: Move constant multipliers outside the $\int$ sign.
• 3. Eliminating the Original Variable
  • **Leftover $x's**: Sometimes substitution leaves extra x$ terms.
  • Solve for $x$: Rearrange the $u = ...$ equation to find $x$ in terms of $u$.
• 4. Definite Integrals
  • Mapping limits: Evaluate $u(a)$ and $u(b)$ to find new integration bounds.
  • Efficiency: With new limits, you never have to substitute back to $x$.

Visual Anchors

1. The $u$-Substitution Decision Process

2. Transforming Limits of Integration

Definition-Example Pairs

• $u$-Substitution
  • Definition: An integration technique replacing a function $g(x) and its derivative with a single variable u$ and $du$.
  • Real-World Example: Like translating a document from English to Spanish to make it easier for a Spanish speaker to summarize, then translating the summary back to English.
• Definite Integral Bound Change
  • Definition: Recalculating the start and end points of an area calculation to match a new variable framework.
  • Real-World Example: If you switch your speedometer from miles to kilometers, you must also change your trip's start and end markers from miles to kilometers.
• Constant Alteration
  • Definition: Multiplying or dividing the $du$ equation by a constant to precisely match the remaining integrand.
  • Real-World Example: Adjusting a recipe. If a recipe calls for 2 eggs but you only have 1, you scale the entire recipe by a factor of $1/2$.

Worked Examples

▶Example 1: Substitution with Constant Alteration

Problem: Evaluate $\int x(x^2 + 1)^4 \, dx$

Step-by-Step Breakdown:

1. Choose $u$: Let $u = x^2 + 1$ (the "inner" function).
2. Find $du$: $du = 2x \, dx$.
3. Alteration: The integrand has $x \, dx$, not $2x \, dx$. So, divide by 2: $\frac{1}{2}du = x \, dx$.
4. Substitute: $\int u^4 \left(\frac{1}{2}du\right) = \frac{1}{2}\int u^4 \, du$.
5. Integrate: $\frac{1}{2} \cdot \frac{u^5}{5} + C = \frac{u^5}{10} + C$.
6. Back-Substitute: $\frac{(x^2 + 1)^5}{10} + C$.

▶Example 2: Eliminating the Original Variable

Problem: Evaluate $\int x \sqrt{x - 1} \, dx$

Step-by-Step Breakdown:

1. Choose $u$: Let $u = x - 1$.
2. Find $du$: $du = dx$.
3. Handle Leftovers: We still have an $x outside the root. Solve our u$ equation for $x$: $x = u + 1$.
4. Substitute Everything: $\int (u + 1)\sqrt{u} \, du = \int (u + 1)u^{1/2} \, du$.
5. Distribute: $\int (u^{3/2} + u^{1/2}) \, du$.
6. Integrate: $\frac{2}{5}u^{5/2} + \frac{2}{3}u^{3/2} + C$.
7. Back-Substitute: $\frac{2}{5}(x - 1)^{5/2} + \frac{2}{3}(x - 1)^{3/2} + C$.

▶Example 3: Changing Definite Integral Bounds

Problem: Evaluate $\int_{0}^{1} 2x e^{x^2} \, dx$

Step-by-Step Breakdown:

1. Choose $u$: Let $u = x^2$.
2. Find $du$: $du = 2x \, dx$. (Perfect match!)
3. Change Bounds:
   • Lower bound: $x = 0 \implies u = 0^2 = 0$.
   • Upper bound: $x = 1 \implies u = 1^2 = 1$.
4. Substitute: $\int_{0}^{1} e^u \, du$. (Note: these are $u$-bounds now!)
5. Integrate & Evaluate: $[e^u]_0^1 = e^1 - e^0 = e - 1$.

Checkpoint Questions

1. Why must we sometimes solve for $x$ in terms of $u$ after making our initial substitution?
2. When altering the $du$ equation, why are we only allowed to multiply or divide by constants (and not variables)?
3. If you perform a $u-substitution on a definite integral and update your limits of integration, do you need to back-substitute x$ at the very end? Why or why not?
4. What is the fundamental difference between evaluating an indefinite integral using substitution versus a definite integral?

[!TIP] Check your answers: Try solving the Checkpoint Questions without looking back at the text. If you're stuck on question 3, review the "Definite Integrals" section in the Hierarchical Outline!