Lecture on Integration and Antiderivatives

Introduction to Integration

• Integration: Finding the antiderivative of functions.
• Power Rule for Derivatives: Derivative of (x^n) is (n \cdot x^{n-1}).
• Power Rule for Integration (Antiderivatives):
  • Add 1 to the exponent.
  • Divide by the new exponent.
  • Add the constant of integration (C).

Examples

Example 1: Power Rule

• Derivative of (x^3):
  • (3x^2)
• Antiderivative of (3x^2):
  • (x^3 + C)

Example 2: More on Antiderivatives

• Antiderivative of (x^4): (\frac{1}{5}x^5 + C)
• Antiderivative of (x^2): (\frac{1}{3}x^3 + C)
• Antiderivative of (x^7): (\frac{1}{8}x^8 + C)
• Antiderivative of (x): (\frac{1}{2}x^2 + C)

Fractions and Constants

• Antiderivative of (x/4): (\frac{1}{8}x^2 + C)
• Antiderivative of a Constant (e.g., 4): (4x + C)

Integration of Binomials and Trinomials

• Example: (7x - 6)
  • Antiderivative: (\frac{7}{2}x^2 - 6x + C)
• Example: (6x^2 + 4x - 7)
  • Antiderivative: (2x^3 + 2x^2 - 7x + C)

Integration of Radical Functions

• Antiderivative of (\sqrt{x}):
  • Rewrite as (x^{1/2}).
  • Result: (\frac{2}{3}x^{3/2} + C).
• Example: Cube root transformations and integration.

Trigonometric Integrals

• Common Integrals:
  • (\int \cos x, dx = \sin x + C)
  • (\int \sin x, dx = -\cos x + C)
  • (\int \sec^2 x, dx = \tan x + C)
  • Etc.

Indefinite vs Definite Integrals

• Indefinite Integrals: Result in functions with (C).
• Definite Integrals: Result in numerical values without (C).

Fundamental Theorem of Calculus

• Definite Integral: (\int_a^b f(x), dx = F(b) - F(a))

Integration of Exponential Functions

• Antiderivative of (e^{ax}): (\frac{1}{a}e^{ax} + C)

U-Substitution Technique

• Useful for solving integrals of functions where direct integration is complex.
• Example: (e^{8x}) becomes (\frac{1}{8}e^{8x} + C) using substitution.

Integration of Rational Functions

• Example: (1/x^2) becomes (-1/x + C).
• Using U-Substitution: Simplifies complex rational expressions.

Special Cases

• Antiderivative of (1/x): (\ln|x| + C).
• Example with U-Substitution: Simplifies transformations in complex logarithmic integrals.

Conclusion

• Emphasis on practice and familiarization with rules and substitution techniques.
• Knowledge of trigonometric, exponential, and logarithmic integrals is crucial.