Indefinite Integration Lecture Notes

Introduction

• Discussed indefinite integrals and anti-derivatives.
• Emphasized the necessity of adding a constant C when finding an antiderivative.

Anti-derivatives of Constants

• Integral of 4 dx:
  • Result: 4x + C
• Integral of pi dy:
  • Result: πy + C
• Integral of e dz:
  • Result: ez + C

Variable Raised to Constant

• Formula:
  • ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
• Example:
  • ∫x² dx = x³/3 + C
• Example:
  • ∫8x³ dx = 2x⁴ + C

Polynomial Functions

• Integrate each term separately.
• Example: x² - 5x + 6
  • Result: x³/3 - 5x²/2 + 6x + C

Square Root Functions

• Rewrite using fractional exponents.
• Example: ∫√x dx = 2/3 x^(3/2) + C

U-Substitution

• Technique used for substitution to simplify integration.
• Example: ∫x² sin(x³) dx
  • Let u = x³ and differentiate to find du

Integration by Parts

• Formula: ∫u dv = uv - ∫v du
• Apply when the product of functions cannot be easily integrated using substitution.
• Example: ∫x e^(4x) dx
  • Let u = x and dv = e^(4x) dx

Trigonometric Integrals

• Example: ∫cos(x) dx = sin(x) + C
• Example: ∫sin(x) dx = -cos(x) + C

Exponential Functions

• Example: ∫e^(4x) dx = e^(4x)/4 + C

Special Techniques

• Trigonometric Substitution used for integrals involving √(1 + x²) or √(1 - x²).
• Inverse trigonometric functions used with substitution for expressions like 1/(1 + x²).

Summary

• Reviewed multiple integration techniques including power rule, substitution, integration by parts, and trigonometric substitution.
• Emphasized transformation of integrals to more workable forms.