Basic Rules for Finding Indefinite Integrals

1. Power Rule:

• Formula:${x}^{n}$ dx = $\frac{{x}^{\left(n+1\right)}}{\left(n+1\right)}$ + C (except when n = -1)
• Explanation: Integrating a power function involves increasing the exponent by 1 and dividing by the new exponent.
• Examples:
• ${x}^{4}$ dx = $\frac{{x}^{4}}{4}$ + C
• ${x}^{\left(-2\right)}$ dx = -${x}^{\left(-1\right)}$ + 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:
• ∫ (3${x}^{2}$ + 4x) dx = ∫ 3${x}^{2}$ dx + ∫ 4x dx = ${x}^{3}$ + 2${x}^{2}$ + 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 = ${x}^{2}$+ 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 = ${x}^{2}$ + C

6. Exponential and Logarithmic Rules:

• Exponential Function:${e}^{x}$ dx = ${e}^{x}$ + C
• Natural Logarithm:$\frac{1}{x}$ 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.