Antiderivatives and Integration

Overview

• Antiderivative: The opposite of differentiation.
  • If the derivative of f(x) is f'(x), then the antiderivative (integral) of f'(x) is f(x).
• Notation:
  • F(x) is the antiderivative of f(x).
  • f(x) is the antiderivative of f'(x).
• Process:
  • Integration: Finding the antiderivative.
  • Differentiation: Finding the derivative.

Types of Integrals

• Indefinite Integral:
  • Gives a function, e.g., ∫f(x)dx = F(x) + C, where C is the constant of integration.
• Definite Integral:
  • Results in a number and has limits of integration.

Power Rule for Antiderivatives

• Derivative of x^n: nx^(n-1).
• Antiderivative of x^n: x^(n+1)/(n+1) + C.

Examples

• Antiderivative of 3x²:
  • ∫3x² dx = 3x³/3 + C = x³ + C.
• Antiderivative of Polynomial:
  • f(x) = x³ - 4x² + 8x
  • ∫(x³ - 4x² + 8x)dx = x⁴/4 - (4x³/3) + 4x² + C

Finding Antiderivatives

• Constant Integration:
  • ∫5 dx = 5x + C
• Rational Functions:
  • ∫1/x dx = ln|x| + C.

Integration Techniques

• Integration of Constants:
  • Example: ∫8 dr = 8r + C.
• Using Power Rule for Negative Exponents:
  • Example: ∫1/x³ dx = -1/(2x²) + C.

Complex Examples

• Integration of Sum of Terms:
  • ∫(5x⁷ - 9/x² + 4x - 8) dx
  • = (5x⁸/8) + (9/x) + 2x² - 8x + C

Special Integrals

• Logarithmic Form:
  • ∫1/(ax + b) dx = (1/a)ln|ax + b| + C
  • Example: ∫1/(3x + 4) dx = (1/3)ln|3x + 4| + C

Important Concepts

• Absolute Value in Logs:
  • Necessary for expressions that could be negative.
• Derivatives of Natural Logs:
  • d/dx [ln(u)] = u'/u

Additional Notes

• Function of a Constant:
  • ∫π dθ = πθ + C
• Exponential Integrals:
  • ∫e ds = es + C
• Integration of Rational Expressions:
  • Examples shown for various forms, including transformations using u substitution.