Antiderivatives and indefinite integrals

Section 5.1 Antiderivatives and indefinite integrals

We are now masters of differentiation. Given a function, we know how to calculate its derivative. Great! But what about the inverse process? Suppose that you are given a function

f (x) :

$f(x)\text{:}$ what other functions

F (x)

$F(x)$ are such that their derivatives

F^{'} (x)

$F'(x)$ are equal to the original function

f (x) ?

$f(x)\text{?}$ From the point of view of kinematics: suppose that you know the velocity function of an object, can you determine its position function? From the point of view of geometry: suppose that you know the slope of the tangent lines to the graph of a function at all points, can you determine what the original function is?¹

The answer to these last two questions is: almost. As we will see, you would also need one more piece information to answer unambiguously these questions: either the position of the object at a certain time, or the coordinates of a point that the graph of the function passes through. This would unambiguously fix the appropriate antiderivative, or, in other words, fix the “constant of integration”, as we will see.

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This is the idea behind integration. In this section we take the first steps. We define the concept of “antiderivatives”: an antiderivative of a function

$f$ is another function

$F$ such that its derivative is equal to the original function

$f\text{.}$ We also introduce the idea of “indefinite integral”, denoted by

$\displaystyle \int f(x)\ dx\text{,}$ which represents the most general antiderivative of a function

$f\text{.}$

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Objectives

You should be able to:

Explain the meaning of an antiderivative and an indefinite integral.
Use correct notation for antiderivatives and indefinite integrals.
Recall the antiderivatives of elementary functions.
Evaluate indefinite integrals of simple functions.

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Subsection 5.1.1 Instructional video

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Subsection 5.1.2 Key concepts

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Concept 5.1.1. Antiderivatives.

A function $F$ is called an antiderivative of $f$ on an interval $I$ if $F'(x) = f(x)$ for all $x \in I\text{.}$

If $F$ is an antiderivative of $f$ on $I\text{,}$ then the most general antiderivative of $f$ on $I$ is

$\begin{equation*} F(x) + C, \end{equation*}$

where $C \in \mathbb{R}$ is an arbitrary constant.

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Concept 5.1.2. Indefinite integrals.

We use the notation

$\begin{equation*} \int f(x)\ dx = F(x) + C \end{equation*}$

to denote the most general antiderivative of $f\text{;}$ this expression is called the indefinite integral of $f$ .

The notation $\displaystyle \int f(x) \ dx$ means “finding the general antiderivative of $f(x)\text{,}$ ” just as $\displaystyle \frac{d}{dx} f(x)$ means “finding the derivative of $f(x)\text{.}$ ”

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Concept 5.1.3. Table of indefinite integrals.

First, we note the following two important properties of indefinite integrals:

$\begin{align*} \int c\ f(x)\ dx =\amp c \int f(x)\ dx \qquad \text{for $c$ a constant},\\ \int \left(f(x) + g(x) \right)\ dx =\amp \int f(x)\ dx + \int g(x)\ dx. \end{align*}$

Next, we list below a few well known indefinite integrals. These formulae can be verified by differentiating the right-hand-side and obtaining the integrand on the left-hand-side (the thing inside the indefinite integral, without the $dx$ ).

$\begin{align*} \int x^n\ dx =\amp \frac{1}{n+1} x^{n+1} + C \quad (n \neq -1)\\ \int \frac{1}{x}\ dx =\amp \ln|x| + C\\ \int e^x\ dx =\amp e^x + C\\ \int a^x\ dx =\amp \frac{a^x}{\ln(a)} + C\\ \int \sin x\ dx=\amp -\cos x+ C\\ \int \cos x\ dx =\amp \sin x + C\\ \int \sec^2 x\ dx =\amp \tan x + C\\ \int \csc^2 x \ dx =\amp - \cot x+ C\\ \int \sec x \tan x \ dx =\amp \sec x + C\\ \int \csc x \cot x \ dx =\amp - \csc x + C\\ \int \frac{1}{x^2+1}\ dx =\amp \tan^{-1} x + C\\ \int \frac{1}{\sqrt{1-x^2}}\ dx =\amp \sin^{-1} x + C\\ \int \frac{1}{x \sqrt{x^2-1}}\ dx =\amp \sec^{-1} x + C \end{align*}$

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MATH 144: Calculus for the Physical Sciences I