Introduction To Integrals: Antiderivatives Rules, How To Find Antiderivatives Using Reverse Rules

Learning Objectives

Find the general antiderivative of a given function.Explain the terms and notation used for an indefinite integral.State the power rule for integrals.Use antidifferentiation to solve simple initial-value problems.

Đang xem: Antiderivatives rules

At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function how do we find a function with the derivative and why would we be interested in such a function?

We answer the first part of this question by defining antiderivatives. The antiderivative of a function is a function with a derivative Why are we interested in antiderivatives? The need for antiderivatives arises in many situations, and we look at various examples throughout the remainder of the text. Here we examine one specific example that involves rectilinear motion. In our examination in Derivatives of rectilinear motion, we showed that given a position function of an object, then its velocity function is the derivative of —that is,

*

Furthermore, the acceleration

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is the derivative of the velocity —that is,

*

Now suppose we are given an acceleration function

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but not the velocity function

*

or the position function

*

Since

*

determining the velocity function requires us to find an antiderivative of the acceleration function. Then, since

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determining the position function requires us to find an antiderivative of the velocity function. Rectilinear motion is just one case in which the need for antiderivatives arises. We will see many more examples throughout the remainder of the text. For now, let’s look at the terminology and notation for antiderivatives, and determine the antiderivatives for several types of functions. We examine various techniques for finding antiderivatives of more complicated functions in the second volume of this text (Introduction to Techniques of Integration).

The Reverse of Differentiation

At this point, we know how to find derivatives of various functions. We now ask the opposite question. Given a function how can we find a function with derivative If we can find a function derivative we call an antiderivative of

Definition

A function is an antiderivative of the function if

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for all in the domain of

Consider the function

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Knowing the power rule of differentiation, we conclude that

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is an antiderivative of since

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Are there any other antiderivatives of Yes; since the derivative of any constant is zero, is also an antiderivative of

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Therefore,

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and

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are also antiderivatives. Are there any others that are not of the form for some constant

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The answer is no. From Corollary 2 of the Mean Value Theorem, we know that if and are differentiable functions such that

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then

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for some constant

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This fact leads to the following important theorem.

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General Form of an Antiderivative

Let be an antiderivative of over an interval Then,

for each constant the function is also an antiderivative of over

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if is an antiderivative of over

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there is a constant for which

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over

In other words, the most general form of the antiderivative of over

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is

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We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.

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For each of the following functions, find all antiderivatives.

*
*
*
*

Show Answera. Because

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then

*

is an antiderivative of Therefore, every antiderivative of

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is of the form for some constant and every function of the form is an antiderivative of

b. Let

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For

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0,f(x)=\text{ln}(x)” title=”Rendered by QuickLaTeX.com” height=”18″ width=”146″ style=”vertical-align: -4px;” /> and

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For

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*

Hint

What function has a derivative of

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Indefinite Integrals

We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function we use the notation

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or

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to denote the derivative of Here we introduce notation for antiderivatives. If is an antiderivative of we say that is the most general antiderivative of and write

The symbol

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is called an integral sign, and

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is called the indefinite integral of

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if and only if is an antiderivative of Therefore, when claiming that

Verifying an Indefinite Integral

Each of the following statements is of the form Verify that each statement is correct by showing that

*

*

the statement

is correct.Note that we are verifying an indefinite integral for a sum. Furthermore,

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and

*

are antiderivatives of and

*

respectively, and the sum of the antiderivatives is an antiderivative of the sum. We discuss this fact again later in this section.Using the product rule, we see that
Therefore, the statement

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