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 $f(x)\text{:}$ what other functions $F(x)$ are such that their derivatives $F'(x)$ are equal to the original function $f(x)\text{?}$ 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.

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 $f\text{.}$

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.

Subsection 5.1.1 Instructional video

Subsection 5.1.2 Key concepts

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.

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 $f(x)\text{.}$”

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*}

MATH 144: Calculus for the Physical Sciences I