Module 5: Anti-Differentiation by Substitution

Learning Objective

• Compute the antiderivative of a function using the substitution rule.

Recap: Antiderivatives of Trigonometric Functions

• Antiderivative of sine x dx:
  • (-\cos x + C)
• Antiderivative of cosine x dx:
  • (\sin x + C)
• Antiderivative of sec^2 x dx:
  • (\tan x + C)
• Antiderivative of csc^2 x dx:
  • (-\cot x + C)
• Antiderivative of sec x tan x dx:
  • (\sec x + C)
• Antiderivative of csc x cot x dx:
  • (-\csc x + C)

Introduction to Integration by Substitution

• Also known as u-substitution.
• Useful for integrals in the form:
  • (\int f(g(x)) \cdot g'(x) , dx)
• Identify (g(x)) and its derivative (g'(x) dx).

Examples

Example 1

• Problem: (\int (x-1)^3 , dx)
• Method:
  • Let (u = x - 1), hence (du = dx).
  • Rewrite integral: (\int u^3 , du)
  • Integrate: (\frac{u^4}{4} + C)
  • Substitute back: (\frac{(x-1)^4}{4} + C)

Example 2

• Problem: (\int (x^5 - 3)^8 \cdot 5x^4 , dx)
• Method:
  • Let (u = x^5 - 3), hence (du = 5x^4 , dx).
  • Rewrite integral: (\int u^8 , du)
  • Integrate: (\frac{u^9}{9} + C)
  • Substitute back: (\frac{(x^5 - 3)^9}{9} + C)

Example 3

• Problem: (\int (5x - 2)^7 , dx)
• Method:
  • Let (u = 5x - 2), hence (du = 5 , dx).
  • (\frac{1}{5} du = dx)
  • Rewrite integral: (\frac{1}{5} \int u^7 , du)
  • Integrate: (\frac{1}{5} \cdot \frac{u^8}{8} + C)
  • Simplify: (\frac{(5x - 2)^8}{40} + C)

Example 4

• Problem: (\int \frac{2x}{\sqrt{x^2 + 1}} , dx)
• Method:
  • Let (u = x^2 + 1), hence (du = 2x , dx).
  • Rewrite integral: (\int \frac{1}{\sqrt{u}} , du)
  • Integrate: (2\sqrt{u} + C)
  • Substitute back: (2\sqrt{x^2 + 1} + C)

Conclusion

• Practice through activities provided in the module.

Motivation

• Quote by Gordon B. Hinckley: "There is no substitute under the heavens for productive labor..."

• Note: Ensure to work through the practice problems in the module to solidify understanding.