Indefinite Integrals

Antiderivatives Overview

Definition

  • An antiderivative of a function f(x) is a function F such that F'(x) = f(x).

Example

  • To find F such that F' = 3x², one can conclude F = x³ + C, where C is a constant.

Families of Antiderivatives

General Antiderivative

  • F(x) is referred to as an antiderivative of f, not the antiderivative.

  • There are infinitely many antiderivatives for f(x) = 3x², all of the form F(x) = x³ + C.

Theorem 4.1

  • The entire family of antiderivatives can be represented by adding a constant to a known antiderivative.

  • Example: The derivatives of G(x) = x² + C represent all antiderivatives of f(x) = 2x.

Differential Equations

Definition

  • A differential equation includes x, y, and derivatives of y.

  • Examples: y' = 3x and y' = x² + 1.

General Solution Example

  • For y' = 2, the function whose derivative is 2 is y = 2x.

  • The general solution is y = 2x + C.

Notation for Antiderivatives

Antidifferentiation

  • Finding solutions to differential equations is called antidifferentiation (or indefinite integration), denoted by an integral sign ∫.

  • The notation ∫f(x)dx represents the antiderivative of f with respect to x.

Indefinite Integral

  • The term "indefinite integral" is synonymous with "antiderivative."

Basic Integration Rules

Inverse Relationship

  • The concepts of integration and differentiation are inversely related.

  • If ∫f(x)dx = F(x) + C, then F'(x) = f(x).

Rules Overview

  • Basic integration formulas can be derived from basic differentiation formulas.

Applying Basic Integration Rules

Example of Antiderivatives

  • The antiderivatives of 3x can be expressed as F(x) = (3/2)x² + C, where C is any constant.

Initial Conditions and Particular Solutions

General Solutions

  • The equation y = ∫f(x)dx has multiple solutions differing by a constant.

  • Graphs of two antiderivatives of f are vertical translations of each other.

Specific Example

  • Given a general solution F(x) = x³ - x + C, to find a particular solution use initial conditions such as F(2) = 4.

  • Solving for C provides F(x) = x³ - x - 2.

Finding Particular Solutions

Process

  • To find the general solution from a differential equation, integrate.

  • Apply initial conditions to find the specific solution.

  • Example provided: F(1) = 0 leads to a particular solution.