Indefinite Integrals

Subsection 1.5.1 Defining the Indefinite Integral

Recall the definition of the antiderivative from Section 1.1: A function \(F\) is an antiderivative of \(f\) on an interval \(I\) if \(F'(x)=f(x)\) for all \(x\) in \(I\text{.}\) Let us explore the antiderivative concretely by letting \(f(x)=2x\text{.}\) Then we can readily determine that the antiderivative of \(f\) is the function \(F(x)=x^2\text{,}\) i.e. \(F'(x)=f(x)\text{.}\) However, the function \(F(x)+1 = x^2+1\) also has \(f\) as its derivative:

\begin{equation*} \frac{d}{dx}\left(F(x)+1\right) = \frac{d}{dx} \left(x^2+1\right) = 2x\text{.} \end{equation*}

In fact, any function \(F(x)+C=x^2+C\) for some real constant \(C\) has \(f\) as its derivative:

\begin{equation*} \frac{d}{dx}\left(F(x)+C\right) = \frac{d}{dx} \left(x^2+C\right) = 2x\text{.} \end{equation*}

It comes as no surprise to us that the graphs of the family of functions \(F(x)+C\) are visually just vertical displacements of \(F(x)\text{.}\) In the particular case when \(F(x)=x^2\text{,}\) we can also see with the graphs of the family of functions \(F(x)+C\) below that at any point \(x\) the tangent lines are parallel, i.e. the tangent slopes are the same, i.e. the family of functions has the same derivative \(f(x)=2x\text{.}\)

Interactive Demonstration. Drag any point to investigate the tangent lines of the function family \(F(x) = x^2 + C \text{.}\)

This leads us to the following result:

Definition 1.28.

{General Antiderivative} If a function \(F\) is an antiderivative of \(f\) on an interval \(I\text{,}\) then the most general antiderivative of \(f\) on an interval \(I\) is

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

where \(C\) is any real constant.

Let us now define the indefinite integral.

Definition 1.29. The Indefinite Integral.

The set of all antiderivatives of a function \(f(x)\) is the indefinite integral of \(f(x)\) with respect to \(x\) and denoted by

\begin{equation*} \int f(x)\, dx\text{,} \end{equation*}

where

\(\ds\int f(x)\, dx\) is read “the integral of \(f\) w.r.t. \(x\)” ,

the symbol \(\ds \int\) is called the integral sign,

the function \(f\) is referred to as the integrand of the integral, and

the variable \(x\) is called the variable of integration.

Note:

1. The process of finding the indefinite integral is also called integration or integrating \(f(x)\text{.}\)

2. The above definition says that if a function \(F\) is an antiderivative of \(f\text{,}\) then

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

   for some real constant \(C\text{.}\)

3. Unlike the definite integral, the indefinite integral is a function.