Lecture on Integration

Introduction to Integration

• Focus: Finding the antiderivative of functions.
• Key principle: Power Rule for Integration.

Power Rule (Anti-Derivative)

• Power Rule for Derivatives:
  • If f(x) = x^n, then f'(x) = nx^(n-1).
• Power Rule for Integration:
  • If F'(x) = x^n, then F(x) = (x^(n+1))/(n+1) + C.
  • Key difference: Instead of subtracting the exponent by 1, add 1 to the exponent and divide by the new exponent. Add a constant of integration C.

Examples Using Power Rule

1. Finding Antiderivative of 3x^2:
   • 3x^2 → x^3 + C
2. Finding Antiderivative of x^4:
   • x^4 → (x^5)/5 + C
   • Simplified: 1/5 x^5 + C
3. Antiderivatives of x^2 and x^7:
   • x^2 → (x^3)/3 + C
   • x^7 → (x^8)/8 + C

Special Cases

• Antiderivative of x:
  • x → (x^2)/2 + C
• Antiderivative of Fraction (x/4):
  • x/4 → 1/4 * (x^2)/2 + C → (x^2)/8 + C
• **Antiderivative of Polynomial with Constants: `8x^3 **
  • 8x^3 → 2x^4 + C
• Antiderivative of Constants (4, -7):
  • 4 → 4x + C
  • -7 → -7x + C

Binomials and Trinomials

• Example Binomial: 7x - 6:
  • 7x - 6 → (7x^2)/2 - 6x + C
• Example Trinomial: 6x^2 + 4x - 7:
  • 6x^2 + 4x - 7 → 2x^3 + 2x^2 - 7x + C

Radical Functions

• Antiderivative of √x:
  • Rewrite as x^(1/2).
  • (x^(3/2)) * (2/3) + C
• Antiderivative of Cube Root:
  • x^(4/3) → (x^(7/3)) * (3/7) + C
• Fourth Root:
  • x^(7/4) → (x^(11/4)) * (4/11) + C

Trigonometric Functions

• Common Derivatives and Their Antiderivatives:
  • d/dx [sin x] = cos x → ∫ cos x dx = sin x + C
  • d/dx [cos x] = -sin x → ∫ -sin x dx = cos x + C
  • d/dx [tan x] = sec^2 x→ ∫ sec^2 x dx = tan x + C
  • d/dx [cot x] = -csc^2 x → ∫ -csc^2 dx = cot x + C
  • d/dx [sec x] = sec x tan x → ∫ sec x tan x dx = sec x + C
  • d/dx [csc x] = -csc x cot x → ∫ csc x cot x dx = -csc x + C
• Example:
  • 4 sin x - 5 cos x + 3 sec^2 x → -4 cos x - 5 sin x + 3 tan x + C

Indefinite vs. Definite Integrals

• Indefinite Integral: Results in a function of x, plus C.
• Definite Integral: Calculated with bounds; results in a numerical value.
• Example:
  • ∫ 6x^2 dx → 2x^3 + C
  • ∫ from 1 to 2 of 6x^2 dx → 14
  • Implemented using the Fundamental Theorem of Calculus:
    • ∫ from a to b f(x) dx = F(b) - F(a)

Exponential Functions

• Derivative and Antiderivative:
  • d/dx [e^u] = e^u * u'
  • ∫ e^u du = e^u / u' + C
• Examples:
  • ∫ e^x dx = e^x + C
  • ∫ e^5x dx = (e^5x)/5 + C
  • Using substitution:
    • 4x e^(x^2) dx → 2 e^(x^2) + C

Rational Functions and U-substitution

• Example Rational Anti-derivatives:
  • ∫ (1/x^2) dx → -1/x + C
  • ∫ (1/x^3) dx → - (1/2x^2) + C
  • ∫ (8/x^4) dx → - (8/3x^3) + C
• Using U-substitution:
  • ∫ (1/(4x - 3)^2) dx → - (1/(4(4x - 3))) + C
  • ∫ (7/(5x - 3)^4) dx → - (7/(15(5x - 3)^3)) + C

Natural Logarithms (ln)

• Antiderivative of 1/x:
  • ∫ (1/x) dx = ln|x| + C
• Examples:
  • ∫ (7/x) dx = 7 ln|x| + C
  • ∫ (1/(x + 5)) dx = ln|x + 5| + C
  • Using substitution: Get same results as direct methods.

Summary

• Practice antiderivatives using rules and substitutions.
• Apply knowledge to polynomials, radicals, exponentials, and trig functions.