# 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

is the derivative of the velocity —that is,

Now suppose we are given an acceleration function

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

See also  Math Sc E Derivative Of Chain Rule (Video), Simple Examples Of Using The Chain Rule

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

for all in the domain of

Consider the function

Knowing the power rule of differentiation, we conclude that

is an antiderivative of since

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

Therefore,

and

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

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

then

for some constant

This fact leads to the following important theorem.

### 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

if is an antiderivative of over

there is a constant for which

over

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

is

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

For each of the following functions, find all antiderivatives.

then

is an antiderivative of Therefore, every antiderivative of

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

b. Let

For

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

For

Hint

What function has a derivative of

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

or

See also  3 Derivatives Of Hollandaise Sauce And Its Derivatives, Hollandaise Derivative Sauces

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

is called an integral sign, and

is called the indefinite integral of

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,

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

See more articles in category: Derivative