Chapter 4: Integration

4.1 Antiderivatives and Indefinite Integration

Antiderivatives and indefinite integration form the foundation of integral calculus. This section introduces the concept of finding a function whose derivative is a given function, explores the notation and properties of indefinite integrals, and demonstrates how to solve basic differential equations.

Definition of Antiderivative

• Antiderivative: A function F is an antiderivative of f on an interval I if for all in I.

• There are infinitely many antiderivatives for a given function, differing by a constant.

• The general form: If is an antiderivative of , then every antiderivative is , where is a constant.

Theorem 4.1: Representation of Antiderivatives

• If is an antiderivative of on an interval , then any other antiderivative of on is of the form for some constant .

• Proof Outline: The difference between any two antiderivatives is a constant function.

Indefinite Integral Notation

• The operation of finding all antiderivatives is called antidifferentiation or indefinite integration.

• The indefinite integral of with respect to is written as .

• Integrand: The function inside the integral sign.

• Variable of integration: The variable indicated by .

• Constant of integration: The arbitrary constant added to the antiderivative.

Solving Differential Equations

• A differential equation involves derivatives of an unknown function.

• To solve , integrate both sides: .

• The general solution includes the constant ; a particular solution is found by applying an initial condition.

Examples of Antidifferentiation

• For , .

• For , .

• For , , .

Basic Integration Rules

Strategies for Integration

• Rewrite the integrand to fit a basic rule (e.g., use rational exponents, split fractions, or apply trigonometric identities).

• Check your answer by differentiating the result.

Examples

• Example:

• Example:

• Example: can be rewritten using identities before integrating.

Initial Conditions and Particular Solutions

• To find a particular solution, use the initial condition to solve for in .

• All antiderivatives of a function are vertical translations of each other.

• Example: If and , then and the particular solution is .

Applications: Vertical Motion

• Integration is used to solve motion problems where acceleration is known.

• For a ball thrown upward with initial velocity and initial height , under gravity :

• Find the time when the object hits the ground by solving .

Concept Check

• What does it mean for to be an antiderivative of ? for all in the interval.

• Can two different functions both be antiderivatives of the same function? Yes, they differ by a constant.

• How do you find a particular solution? Integrate to find the general solution, then use the initial condition to solve for .

• Difference between general and particular solution: The general solution contains the arbitrary constant ; the particular solution has determined by an initial condition.

Summary Table: Basic Integration Rules

Additional info: The section also includes exercises on verifying integration by differentiation, solving differential equations, rewriting integrals, and applying integration to physical problems such as vertical motion and population growth. Students are encouraged to check their answers by differentiating their results and to use integration as a tool for solving real-world problems involving rates of change.