5 Integration

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5.1 Antiderivatives and Indefinite Integration

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Objectives

• Write the general solution of a differential equation and use indefinite integral notation for antiderivatives.
• Use basic integration rules to find antiderivatives.
• Find a particular solution of a differential equation.

Antiderivatives

Definition and Explanation

• To find a function $F$ whose derivative is $f(x) = 3x^2$, one can conclude that the function $F$ is an antiderivative of $f$.
• Antiderivative: A function $F$ is called an antiderivative of the function $f$ if $F' = f$.

Explanation of the Term

• The function $F$ is referred to as an antiderivative of $f$ rather than the antiderivative of $f$ because multiple functions can have the same derivative.
• For any constant $C$, the function $F(x) = x^3 + C$ is an antiderivative of $f(x) = 3x^2$.
  

Theorem 5.1 (Family of Antiderivatives)

• According to Theorem 5.1, all antiderivatives of a function can be represented by adding a constant to a known antiderivative.
• Example: Since $D[x^2] = 2x$, you can represent the entire family of all antiderivatives of $f(x) = 2x$ by:
  $G(x) = x^2 + C$
    where $C$ is a constant, known as the constant of integration.

General Antiderivative

• The family of functions represented by $G$ is referred to as the general antiderivative of $f$.
• Therefore, the expression $G(x) = x^2 + C$ is the general solution of the differential equation $G'(x) = 2x$.

Differential Equations

• A differential equation is an equation that involves the variable $x$, a function $y$, and derivatives of $y$.
• Example: $y' = 3x$ and $y' = x^2 + 1$ are both instances of differential equations.

Example 1 – Solving a Differential Equation

Problem Statement

• Find the general solution of the differential equation $y' = 2$.

Solution

• To solve, determine a function whose derivative is 2: one such function is $y = 2x$, which is an antiderivative of 2.
• Therefore, employing Theorem 5.1 indicates that the general solution to this differential equation is:
  $y = 2x + C$

Understanding the Operation of Antidifferentiation

Definition

• The operation of finding all solutions to a differential equation is termed antidifferentiation or indefinite integration.
• This is denoted by an integral sign
  ext{∫}.

Notation

• The general solution is denoted by the expression:
  $ext{∫} f(x) ext{d}x$
• This is read as the antiderivative of $f$ with respect to $x$.
• The differential $ext{dx}$ identifies $x$ as the variable of integration.
• The term indefinite integral is a synonym for antiderivative.

Basic Integration Rules

Introduction

• The inverse nature of integration and differentiation can be established by substituting $F'(x)$ for $f(x)$ in the definition of indefinite integration.

Integration Formulas from Differentiation

• From the relationship $ext{∫} f(x) ext{d}x = F(x) + C$, one obtains integration formulas directly derived from differentiation formulas, as shown below:

Summary of Basic Integration Rules

Example 2 – Describing Antiderivatives

• The antiderivatives of $3x$ are of the form:
  $ext{Antiderivative}(3x) = \frac{3x^2}{2} + C.$
• General integration patterns are similar to those of differentiation.

Initial Conditions and Particular Solutions

Explanation

• The equation $y = ext{∫}f(x) ext{d}x$ has numerous solutions differing by a constant value.
• Consequently, the graphs of any two antiderivatives of $f$ result in vertical translations of one another.

Example with Graphs

• For instance, graphs reflecting varying integer values of $C$ reveal the different functions.

Determining Particular Solutions

• In many applications, sufficient information is provided to ascertain a particular solution by knowing the value of $y = F(x)$ for specific $x$ values, referred to as an initial condition.

Example of Initial Condition

• If the general solution is $F(x) = x^3 - x + C$ and given the initial condition $F(2) = 4$, one can derive that:
  $F(2) = 2^3 - 2 + C = 4$, leading to:
    8 - 2 + C = 4
  ightarrow C = -2.
• Thus, the specific solution becomes:
  $F(x) = x^3 - x - 2.$

Example 8 – Finding a Particular Solution

• To find the general solution of $F'(x) = e^x$ and the particular solution satisfying the initial condition $F(0) = 3$:

1. Integrate to obtain the general solution.
2. Use the initial condition $F(0) = 3$ to solve for $C$.
   

Steps to Solve

• Integrate: $F(x) = e^x + C$.
• Solving for C:
    F(0) = e^0 + C
  ightarrow 3 = 1 + C
  ightarrow C = 2.
• Hence, the particular solution is:
  $F(x) = e^x + 2.$
• Note: The solution curves corresponding to values of $C = -3, -2, -1, 0, 1, 2, 3$ are illustrated in the associated figures.