Given a function \(y=f(x)\text{,}\) a differential equation is an equation that incorporates \(y\text{,}\) \(x\text{,}\) and the derivatives of \(y\text{.}\) For instance, a simple differential equation is: \begin{equation*} y' = 2x \text{.} \end{equation*}

Solving a differential equation amounts to finding a function \(y\) that satisfies the given equation. Take a moment and consider that equation; can you find a function \(y\) such that \(y' = 2x\text{?}\)

Can you find another?

And yet another?

Hopefully you were able to come up with at least one solution: \(y = x^2\text{.}\) “Finding another” may have seemed impossible until one realizes that a function like \(y=x^2+1\) also has a derivative of \(2x\text{.}\) Once that discovery is made, finding “yet another” is not difficult; the function \(y = x^2 + 123{,}456{,}789\) also has a derivative of \(2x\text{.}\) The differential equation \(y' = 2x\) has many solutions. This leads us to some definitions.

Make a note about our definition: we refer to an antiderivative of \(f\text{,}\) as opposed to the antiderivative of \(f\text{,}\) since there is always an infinite number of them. We often use upper-case letters to denote antiderivatives.

When \(f\) is continuous, knowing one antiderivative of \(f\) allows us to find infinitely more, simply by adding a constant. Not only does this give us more antiderivatives, it gives us all of them.

Given a continuous function \(f\) and one of its antiderivatives \(F\text{,}\) we know all antiderivatives of \(f\) have the form \(F(x) + C\) for some constant \(C\text{.}\) Using Definition 5.1.1, we can say that \begin{equation*} \int f(x) \,dx = F(x) + C. \end{equation*}

Let's analyze this indefinite integral notation.

Figure 5.1.3 shows the typical notation of the indefinite integral. The integration symbol, \(\int\text{,}\) is in reality an “elongated S,” representing “take the sum.” We will later see how sums and antiderivatives are related.

The function we want to find an antiderivative of is called the integrand. It contains the differential of the variable we are integrating with respect to. The \(\int\) symbol and the differential \(dx\) are not “bookends” with a function sandwiched in between; rather, the symbol \(\int\) means “find all antiderivatives of what follows,” and the function \(f(x)\) and \(dx\) are multiplied together; the \(dx\) does not “just sit there.”

Another important aspect of the \(dx\) is that it tells us which variable we want an antiderivative with respect to. Much like how \(\lzo{x}\) would mean to take the derivative with respect to \(x\text{,}\) while \(\lzo{t}\) would be the derivative with respect to \(t\text{.}\)

Let's practice using this notation.

A commonly asked question is “What happened to the \(dx\text{?}\)” The unenlightened response is “Don't worry about it. It just goes away.” A full understanding includes the following.

This process of antidifferentiation is really solving a differential question. The integral \begin{equation*} \int \sin(x) \,dx \end{equation*} presents us with a differential, \(dy = \sin(x) \ dx\text{.}\) It is asking: “What is \(y\text{?}\)” We found lots of solutions, all of the form \(y = -\cos(x) +C\text{.}\)

Letting \(dy = \sin(x)\,dx\text{,}\) rewrite \begin{equation*} \int \sin(x) \,dx \text{ as } \int\,dy \text{.} \end{equation*}

This is asking: “What functions have a differential of the form \(dy\text{?}\)” The answer is “Functions of the form \(y+C\text{,}\) where \(C\) is a constant.” What is \(y\text{?}\) We have lots of choices, all differing by a constant; the simplest choice is \(y = -\cos(x)\text{.}\)

Understanding all of this is more important later as we try to find antiderivatives of more complicated functions. In this section, we will simply explore the rules of indefinite integration, and one can succeed for now with answering “What happened to the \(dx\text{?}\)” with “It went away.”

Let's practice once more before stating integration rules.

This final step of “verifying our answer” is important both practically and theoretically. In general, taking derivatives is easier than finding antiderivatives so checking our work is easy and vital as we learn.

We also see that taking the derivative of our answer returns the function in the integrand. Thus we can say that: \begin{equation*} \lzoo{x}{\int f(x)\,dx} = f(x). \end{equation*}

Differentiation “undoes” the work done by antidifferentiation.

Theorem 2.7.11 gave a list of the derivatives of common functions we had learned at that point. We restate part of that list here to stress the relationship between derivatives and antiderivatives. This list will also be useful as a glossary of common antiderivatives as we learn.

We highlight a few important points from Theorem 5.1.6.

• \begin{equation*} \int c\cdot f(x)\,dx = c\cdot \int f(x)\,dx \end{equation*}

  This is the Constant Multiple Rule: we can temporarily ignore constants when finding antiderivatives, just as we did when computing derivatives (i.e., \(\lzoo{x}{3x^2}\) is just as easy to compute as \(\lzoo{x}{x^2}\)). An example: \begin{equation*} \int 5\cos(x) \,dx = 5\cdot\int \cos(x) \,dx = 5\cdot (\sin(x) +C) = 5\sin(x) + C. \end{equation*} In the last step we can consider the constant as also being multiplied by 5, but “5 times a constant” is still a constant, so we just write “\(C\)”.

• \begin{equation*} \int \big(f(x)\pm g(x)\big)\,dx =\int f(x)\,dx\pm \int g(x)\,dx \end{equation*}

  This is the Sum/Difference Rule: we can split integrals apart when the integrand contains terms that are added/subtracted, as we did in Example 5.1.5. So: \begin{align*} \int(3x^2+4x+5)\ dx \amp = \int 3x^2\ dx + \int 4x\,dx + \int 5\,dx\\ \amp = 3\int x^2\ dx + 4\int x\,dx + \int 5 \,dx\\ \amp = 3\cdot \frac13x^3 + 4\cdot \frac12x^2+5x+C\\ \amp = x^3+2x^2+5x+C \end{align*} In practice we generally do not write out all these steps, but we demonstrate them here for completeness.

• \begin{equation*} \int x^n\,dx =\frac{1}{n+1}x^{n+1}+ C\quad(n\neq -1) \end{equation*}

  This is the Power Rule of indefinite integration. There are two important things to keep in mind:

  1. Notice the restriction that \(n\neq -1\text{.}\) This is important: \(\int \frac{1}{x}\,dx \neq\) “\(\frac{1}{0}x^0+C\)”; rather, see the last rule from the list.

  2. We are presenting antidifferentiation as the “inverse operation” of differentiation. Here is a useful quote to remember:

     “Inverse operations do the opposite things in the opposite order.”

     When taking a derivative using the Power Rule, we first multiply by the power, then second subtract \(1\) from the power. To find the antiderivative, do the opposite things in the opposite order: first add \(1\) to the power, then second divide by the power.

• \begin{equation*} \int \frac{1}x\,dx = \ln\abs{x}+C \end{equation*}

  Note that this rule uses the absolute value of \(x\text{.}\) The exercises will work the reader through why this is the case; for now, know the absolute value is important and cannot be ignored.