Lecture Notes on Integration

Antiderivatives and Integration Basics

Power Rule for Integration

• Power rule for differentiation: (d/dx)[x^n] = n*x^(n-1)
• Power rule for integration:
  • Add 1 to the exponent: x^(n+1)
  • Divide by the new exponent: (1/(n+1)) * x^(n+1)
  • Add constant of integration, C
• Example:
  • Find the antiderivative of 3x^2:
    • Add 1 to exponent: 2+1 = 3
    • Divide by new exponent: 3x^3/3
    • Add constant: x^3 + C

Examples

• Antiderivative of x^4:
  • Add 1: x^5
  • Divide: x^5/5
  • Add C: (1/5)x^5 + C
• Antiderivative of x^2 and x^7:
  • x^3/3 + C
  • x^8/8 + C
• Antiderivative of x:
  • x^2/2 + C
• Handling fractions and constants:
  • Rewrite and move constants:
  • Example: Integral of x/4 -> 1/4 * Integral of x dx = (1/4)(x^2/2) = x^2/8 + C
• Handling monomials:
  • Example: Integral of 8x^3 -> 2x^4 + C
• Integral of a constant:
  • Example: Integral of 4 dx = 4x + C

Complex Antiderivatives

• Antiderivative of square root of x:
  • Rewrite as x^(1/2): Integral of x^(1/2) dx
  • Apply power rule: (2/3)x^(3/2) + C

Integration of Trigonometric Functions

• Important derivatives and integrals:
  • Integral of cos(x) dx = sin(x) + C
  • Integral of sin(x) dx = -cos(x) + C
  • Integral of sec^2(x) dx = tan(x) + C
  • More to remember:
    • Integral of csc^2(x) dx = -cot(x) + C
    • Integral of sec(x)tan(x) dx = sec(x) + C
    • Integral of csc(x)cot(x) dx = -csc(x) + C
  • Examples:
    • Integral of 4sin(x) - 5cos(x) + 3sec^2(x) dx = -4cos(x) - 5sin(x) + 3tan(x) + C

Indefinite vs. Definite Integrals

• Indefinite Integral:
  • No specified limits
  • Result is a function: ∫6x^2 dx = 2x^3 + C
• Definite Integral:
  • Limits of integration specified: ∫(from A to B) 6x^2 dx
  • Evaluate antiderivative at A and B: [2x^3] from 1 to 2 = 16 - 2 = 14
  • Constant of integration cancels out

Fundamental Theorem of Calculus

• Links differentiation and integration
• ∫(from A to B) f(x) dx = F(B) - F(A), where F(x) is the antiderivative of f(x)

Exponential Functions

• Derivative rule: (d/dx) [e^u] = e^u * u'
• Integral rule for linear u: ∫ e^u du = e^u / u'
  • Examples:
    • ∫ e^(3x) dx = e^(3x)/3 + C
    • ∫ e^(5x) dx = e^(5x)/5 + C

U-Substitution Method

• Used to handle more complex integrands
• Example: ∫4x e^(x^2) dx
  • Let u = x^2, then du = 2x dx
  • Integral becomes ∫2e^u du = 2e^u + C = 2e^(x^2) + C

Antiderivatives of Rational Functions

• Rewriting method: ∫1/x^n dx
  • Example: ∫ 1/x^3 dx
    • Rewrite: x^(-3)
    • Apply power rule: (1/-2)x^(-2) = -1/(2x^2) + C

Integrating Rational Functions Using U-Substitution

• Example: ∫ 1/(4x-3)^2 dx
  • Let u = 4x-3, then du = 4 dx
  • Integral becomes: ∫u^(-2) * (1/4) du = -(4x-3)^(-1)/4 + C

Natural Logarithms

• Special case: ∫ 1/x dx = ln|x| + C
• Examples using constants and shifts:
  • ∫ 7/x dx = 7ln|x| + C
  • ∫ 1/(x+5) dx = ln|x+5| + C