Basic Rules for Finding Indefinite Integrals

1. Power Rule:

  • Formula:xn dx = x(n+1)(n+1) + C (except when n = -1)
  • Explanation: Integrating a power function involves increasing the exponent by 1 and dividing by the new exponent.
  • Examples:
    • x4 dx = x44 + C
    • x(-2) dx = -x(-1) + C

2. Constant Rule:

  • Formula: ∫ k dx = kx + C (where k is a constant)
  • Explanation: Integrating a constant simply involves multiplying the constant by the variable of integration.
  • Examples:
    • ∫ 5 dx = 5x + C
    • ∫ (-2) dx = -2x + C

3. Sum/Difference Rule:

  • Formula: ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
  • Explanation: Integrating a sum or difference of functions can be split into individual integrals.
  • Example:
    • ∫ (3x2 + 4x) dx = ∫ 3x2 dx + ∫ 4x dx = x3 + 2x2 + C

4. Linearity of Integration:

  • Property: ∫[af(x) + bg(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx
  • Explanation: Constants can be factored out of the integral, and the integral distributes over addition and subtraction.
  • Example:
    • ∫ (2x + 3) dx = 2 ∫ x dx + 3 ∫ 1 dx = x2+ 3x + C

5. Constant of Integration:

  • Property: ∫ f(x) dx + C
  • Explanation: Every indefinite integral has a constant of integration (C), representing a family of functions.
  • Example:
    • ∫ 2x dx = x2 + C

6. Exponential and Logarithmic Rules:

  • Exponential Function:ex dx = ex + C
  • Natural Logarithm:1x dx = ln|x| + C
  • Explanation: Integrals of exponential and logarithmic functions have simple forms and are essential in various calculations.

7. Trigonometric Integrals:

  • Sine Function: ∫ sin(x) dx = -cos(x) + C
  • Cosine Function: ∫ cos(x) dx = sin(x) + C
  • Tangent Function: ∫ tan(x) dx = -ln|cos(x)| + C
  • Explanation: Integrals of basic trigonometric functions have specific forms, often requiring knowledge of trigonometric identities.