Overview
This lecture introduces integrals (antiderivatives), explains their basic formulas and properties, and distinguishes between indefinite and definite integrals.
Derivatives and Antiderivatives
- A derivative calculates the rate of change of a function.
- The power rule for derivatives: if ( y = x^n ), then ( y' = n x^{n-1} ).
- An integral is the reverse process of a derivative; it finds the original function from its derivative.
- Integrals are also called antiderivatives.
Indefinite Integrals
- The indefinite integral of ( x^n ) is ( \int x^n dx = \frac{1}{n+1} x^{n+1} + C ) (for ( n \neq -1 )).
- The ( +C ) accounts for the unknown constant lost during differentiation.
- Example: ( \int x^3 dx = \frac{1}{4} x^4 + C ).
- The integral of a sum/difference can be performed on each term separately.
- Constants can be factored out: ( \int k \cdot f(x) dx = k \int f(x) dx ).
- The integral of a constant ( a ) is ( a x + C ).
Definite Integrals
- A definite integral has upper and lower limits: ( \int_a^b f(x) dx ).
- Calculate the antiderivative, then evaluate: ( F(b) - F(a) ).
- The result of a definite integral does not include "+C".
- Properties:
- ( \int_a^a f(x) dx = 0 ).
- Switching the limit order changes the sign: ( \int_a^b f(x) dx = -\int_b^a f(x) dx ).
- Integrals can be split at a point: ( \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx ).
Integral Properties and Operation Rules
- Addition/subtraction of functions: Integrate each term separately.
- Constants outside integrals: Move them in front as multipliers.
- Cannot directly integrate products or quotients unless converted into a sum or using advanced techniques.
- For more complex forms (e.g., products, powers), integration techniques like substitution or partial fractions are needed.
Key Terms & Definitions
- Derivative — Measures how a function changes as its input changes.
- Indefinite Integral — The most general form of the antiderivative, includes "+C".
- Definite Integral — Integral with set upper and lower limits, gives a specific value.
- Antiderivative — Another name for the integral; the original function before differentiation.
- Constant of Integration (C) — Represents all possible constants in the solution to an indefinite integral.
Action Items / Next Steps
- Review the properties and basic formulas for integrals.
- Practice integrating functions using the power rule and combining properties.
- Prepare for the next lesson on integration techniques (e.g., substitution, partial fractions).