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**Copyright © Cengage Learning. All rights reserved.**

4 Integration Copyright © Cengage Learning. All rights reserved.

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**Objectives Write the general solution of a differential equation.**

Use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Find a particular solution of a differential equation.

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**We will also learn some applications.**

Think About It Suppose this is the graph of the derivative of a function What do we know about the original function? Critical numbers Where it is increasing, decreasing What do we not know? f '(x) In this chapter we will learn how to determine the original function when given the derivative. We will also learn some applications. 3

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**Antiderivatives Suppose you were asked to find a function F whose**

derivative is f(x) = 3x2. From your knowledge of derivatives, you would probably say that The function F is an antiderivative of f .

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**Anti-Derivatives Derivatives give us the rate of change of a function**

What if we know the rate of change … Can we find the original function? If F '(x) = f(x) Then F(x) is an antiderivative of f(x) Example – let F(x) = 12x2 Then F '(x) = 24x = f(x) So F(x) = 12x2 is the antiderivative of f(x) = 24x 5

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**Finding An Antiderivative**

Given f(x) = 12x3 What is the antiderivative, F(x)? Use the power rule backwards Recall that for f(x) = xn … f '(x) = n • x n – 1 That is … Multiply the expression by the exponent Decrease exponent by 1 Now do opposite (in opposite order) Increase exponent by 1 Divide expression by new exponent 6

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Example 1 𝑓 ′ 𝑥 = 𝑥 3 Find f(x) = 𝑥 = 𝑥 4 4 = 𝑥 4

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**Example 2: 𝑓 ′ 𝑥 = 𝑥 1 3 Find f(x) = 𝑥 1 3 +1 1 3 +1 = 𝑥 4 3 4 3**

𝑓 ′ 𝑥 = 𝑥 1 3 Find f(x) = 𝑥 = 𝑥 = 𝑥 4 3

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**Example 3 𝑓 ′ 𝑥 = 1 𝑥 4 Find f(x) First Rewrite: 𝑥 −4 = 𝑥 −4+1 −4+1**

𝑓 ′ 𝑥 = 1 𝑥 4 Find f(x) First Rewrite: 𝑥 −4 = 𝑥 −4+1 −4+1 = 𝑥 −3 −3 = −1 3 𝑥 3

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**Example 4 𝑓 ′ 𝑥 = 1 𝑥 1 3 Find f(x) Rewrite first: 𝑥 −1 3**

𝑓 ′ 𝑥 = 1 𝑥 1 3 Find f(x) Rewrite first: 𝑥 −1 3 = 𝑥 − −1 3 +1 = 𝑥 = 𝑥 2 3

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Example 5 𝑓 ′ 𝑥 =2 𝑥 3 Find f(x) = 2 𝑥 = 2 𝑥 4 4 = 𝑥 4

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Example 6 𝑓 ′ 𝑥 =5 Find f(x) = 5 𝑥 = 5𝑥

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**Family of Antiderivatives**

Consider a family of parabolas f(x) = x2 + n which differ only by value of n Note that f '(x) is the same for each version of f Now go the other way … The antiderivative of 2x must be different for each of the original functions So when we take an antiderivative We specify F(x) + C Where C is an arbitrary constant This indicates that multiple antiderivatives could exist from one derivative 13

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**Example y’ = 2x y = (2)(1/2)x1+1 =x2**

what are we missing? Could we have started with y = x or y = x ?? When we find the antiderivative, we need to remember to account for the constant that could have been in the original function. So we get y = x2 + C

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Indefinite Integral The family of antiderivatives of a function f indicated by The symbol is a stylized S to indicate summation, we will discuss this further in later sections of this chapter.

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Integration

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**Indefinite Integral The constant of integration**

The indefinite integral is a family of functions The + C represents an arbitrary constant The constant of integration

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**Integration Example 1 Solve: ∫3𝑥𝑑𝑥 = 3 𝑥 2 2 +𝑐 𝑜𝑟 3 2 𝑥 2 +𝑐 Check:**

= 3 𝑥 𝑐 𝑜𝑟 𝑥 2 +𝑐 Check: 𝑑 𝑑𝑥 𝑥 2 +𝑐 =3𝑥

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**Properties of Indefinite Integrals**

The power rule The integral of a sum (difference) is the sum (difference) of the integrals 19

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**Properties of Indefinite Integrals**

The derivative of the indefinite integral is the original function A constant can be factored out of the integral

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Try It Out Determine the indefinite integrals as specified below

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**Differential Equations**

We will study this topic in chapter 6. For now, here’s a taste of it.

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**Differential Equations**

The family of functions represented by G is the general antiderivative of f, and G(x) = x2 + C is the general solution of the differential equation G'(x) = 2x Differential equation A differential equation in x and y is an equation that involves x, y, and derivatives of y. For instance, y' = 3x and y' = x2 + 1 are examples of differential equations.

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**Example 1 – Solving a Differential Equation**

Find the general solution of the differential equation y' = 2. Solution: To begin, you need to find a function whose derivative is 2. One such function is y = 2x x is an antiderivative of 2. Now, you can use Theorem 4.1 to conclude that the general solution of the differential equation is y = 2x + C General solution

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Example 1 – Solution cont’d The graphs of several functions of the form y = 2x + C are shown in Figure 4.1. Figure 4.1

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**Notation for Antiderivatives**

When solving a differential equation of the form it is convenient to write it in the equivalent differential form The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign ∫.

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**Notation for Antiderivatives**

The general solution is denoted by The expression ∫f(x)dx is read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative.

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**Basic Integration Rules (p.250)**

The inverse nature of integration and differentiation can be verified by substituting F'(x) for f(x) in the indefinite integration definition to obtain Moreover, if ∫f(x)dx = F(x) + C, then

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**Basic Integration Rules**

These two equations allow you to obtain integration formulas directly from differentiation formulas, as shown in the following summary.

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**Basic Integration Rules**

cont’d

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**Basic Integration Rules**

Note that the general pattern of integration is similar to that of differentiation.

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**Example 2 – Applying the Basic Integration Rules**

Describe the antiderivatives of 3x. Solution: So, the antiderivatives of 3x are of the form where C is any constant.

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**Initial Conditions and Particular Solutions**

You have already seen that the equation y = ∫f(x)dx has many solutions (each differing from the others by a constant). This means that the graphs of any two antiderivatives of f are vertical translations of each other.

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**Initial Conditions and Particular Solutions**

For example, Figure 4.2 shows the graphs of several antiderivatives of the form for various integer values of C. Each of these antiderivatives is a solution of the differential equation Figure 4.2

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**Initial Conditions and Particular Solutions**

In many applications of integration, you are given enough information to determine a particular solution. To do this, you need only know the value of y = F(x) for one value of x. This information is called an initial condition. For example, in Figure 4.2, only one curve passes through the point (2, 4). To find this curve, you can use the following information. F(x) = x3 – x + C General solution F(2) = Initial condition

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**Initial Conditions and Particular Solutions**

By using the initial condition in the general solution, you can determine that F(2) = 8 – 2 + C = 4, which implies that C = –2. So, you obtain F(x) = x3 – x – Particular solution

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**Example 7 – Finding a Particular Solution**

Find the general solution of and find the particular solution that satisfies the initial condition F(1) = 0. Solution: To find the general solution, integrate to obtain

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**Example 7 – Solution Using the initial condition F(1) = 0, you**

cont’d Using the initial condition F(1) = 0, you can solve for C as follows. So, the particular solution, as shown in Figure 4.3, is Figure 4.3

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Homework: Pg #2-44E Great to try: p.255#1,4,9,11,13,14,22,31,34,35,38,40,43,47,49,50,59,62,63,65,66,68,71,72,87,90,92,94,95 This is the bare minimum. It is recommended that you do all of the problems in each section to ensure mastery of the concepts. (this includes the word problems!)

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