Deciding Which Integration Technique to Use

Introduction

• Flowchart/Graphic organizer approach
• Overview of types of functions and thought process in choosing techniques

Simple Functions

• Look for functions involving only addition and subtraction of separate pieces
  • Example: x^2, 7x, 1/x, sin(x)
• Integrate each piece individually
  • Example: (1/3)x^3 + 7x^2 + ln(x) - cos(x) + C

Products of Functions

• If functions are multiplied together
  • Example: f(x) * g(x)
• First check for Substitution
  • Look for a part of the function that can be set as u
  • Derivative of u should match another part with a constant multiplier
  • Example: If f(x) = x^2 * sin(x^3), let u = x^3, then du = 3x^2
  • Integrate the transformed function
• If Substitution doesn't work, use Integration by Parts
  • Formula needed: ∫u dv = uv - ∫v du
  • Choose u and dv carefully

Quotients of Functions

• When integrating a function divided by another
  • Example: ∫ f(x) / g(x) dx
• First check for Substitution
  • Set the denominator as u
  • Example: ∫ (3x) / (x^2 + 4) dx, let u = x^2 + 4, then du = 2x
• If Substitution doesn’t work, try Partial Fractions
  • Factor the denominator and decompose
  • Example: (A/x+2) + (B/x+4)
• If Partial Fractions don’t work, use Trigonometric Substitution
  • Look for trigonometric identities to simplify

Division for Simple Quotients

• If the numerator is more complex and the denominator is simple
  • Example: (4x + x^2) / x
  • Simplify to 4 + x and integrate
• If polynomials have the same power, use Polynomial Division
  • Example: (x^2) / (x^2 - 9)
  • Perform division first, then integrate

Special Notes

• Calc AB
  • Responsible for basic substitutions and simple divisions
• Calc BC
  • Responsible for all methods mentioned
• Practice with flowchart and different methods for efficiency

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