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:
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 \(y' = 2x\text{?}\)
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.
Let \(F(x)\) and \(G(x)\) be antiderivatives of a continuous function \(f(x)\) on an interval \(I\text{.}\) Then there exists a constant \(C\) such that, on \(I\text{,}\)
Given a continuous function \(f\) defined on an interval \(I\) and one of its antiderivatives \(F\text{,}\) we know all antiderivatives of \(f\) on \(I\) have the form \(F(x) + C\) for some constant \(C\text{.}\) Using Definition 5.1.2, we can say that
Note that we are abusing notation somewhat: when we write \(F(x)+C\) on the right-hand side, we really mean the set of all such functions, for each real number value of \(C\text{.}\) Let’s analyze this indefinite integral notation.
The equation \(\int f(x) \cdot dx = F(x) + C\text{,}\) with labels for each part. \(\int\) is labeled as the integral symbol. \(f(x)\) is labeled as the integrand function. \(dx\) is labeled as the differential of \(x\text{.}\)\(F(x)\) is labeled as any "antiderivative of \(f\text{.}\)" \(C\) is labeled as the constant of integration.
Figure 5.1.5 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 way of looking at the notation is that it tells us that \(f(x)\,dx\) is the differential of \(F(x)\text{:}\)\(dF(x) = f(x)\,dx\text{,}\) confirming that \(F'(x)=f(x)\text{,}\) as required of an antiderivative. The integral symbol can then be viewed as an instruction to “undo” the differential and recover the antiderivative \(F(x)\text{.}\)
Another important aspect of the \(dx\) is that it tells us which variable we’re taking the 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 \(t\text{.}\)
We are asked to find all functions \(F(x)\) such that \(\Fp(x) = \sin(x)\text{.}\) Some thought will lead us to one solution: \(F(x) = -\cos(x)\text{,}\) because \(\lzoo{x}{-\cos(x)} = \sin(x)\text{.}\)
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.
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 \(y = -\cos(x) +C\text{.}\)
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.”
We seek a function \(F(x)\) whose derivative is \(3x^2+4x+5\text{.}\) When taking derivatives, we can consider functions term-by-term, so we can likely do that here.
What functions have a derivative of \(3x^2\text{?}\) Some thought will lead us to a cubic, specifically \(x^3+C_1\text{,}\) where \(C_1\) is a constant.
What functions have a derivative of \(4x\text{?}\) Here the \(x\) term is raised to the first power, so we likely seek a quadratic. Some thought should lead us to \(2x^2+C_2\text{,}\) where \(C_2\) is a constant.
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.
Theorem 2.7.16 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.
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:
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 5.1.7. So:
This is the Power Rule of indefinite integration. There are two important things to keep in mind:
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.
When taking a derivative using the Power Rule, we firstmultiply by the power, then secondsubtract\(1\) from the power. To find the antiderivative, do the opposite things in the opposite order: firstadd\(1\) to the power, then seconddivide by the power.
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.
In Section 2.3 we saw that the derivative of a position function gave a velocity function, and the derivative of a velocity function describes acceleration. We can now go “the other way:” the antiderivative of an acceleration function gives a velocity function, etc.. While there is just one derivative of a given function, there are infinitely many antiderivatives. Therefore we cannot ask “What is the velocity of an object whose acceleration is -32 ft⁄s2?”, since there is more than one answer.
We can find the answer if we provide more information with the question, as done in the following example. Often the additional information comes in the form of an initial value, a value of the function that one knows beforehand.
The acceleration due to gravity of a falling object is -32 ft⁄s2. At time \(t=3\text{,}\) a falling object had a velocity of -10 ft⁄s. Find the equation of the object’s velocity.
Using the first piece of information, we know that \(v(t)\) is an antiderivative of \(v'(t)=-32\text{.}\) So we begin by finding the indefinite integral of \(-32\text{:}\)
Thus \(v(t)= -32t+86\text{.}\) We can use this equation to understand the motion of the object: when \(t=0\text{,}\) the object had a velocity of \(v(0) = 86\)ft⁄s. Since the velocity is positive, the object was moving upward.
Recognize that we are able to determine quite a bit about the path of the object knowing just its acceleration and its velocity at a single point in time.
Using the initial value, we have found \(\fp(t) = \sin(t) + 3\text{.}\) We now find \(f(t)\) by integrating again. We will use a different integration constant since we have already defined \(C\) to equal \(3\) above.
This section introduced antiderivatives and the indefinite integral. We found they are needed when finding a function given information about its derivative(s). For instance, we found a velocity function given an acceleration function.
In the next section, we will see how position and velocity are unexpectedly related by the areas of certain regions on a graph of the velocity function. Then, in Section 5.4, we will see how areas and antiderivatives are closely tied together. This connection is incredibly important, as indicated by the name of the theorem that describes it: The Fundamental Theorem of Calculus.
If \(F(x)\) is an antiderivative of \(f(x)\text{,}\) and \(G(x)\) is an antiderivative of \(g(x)\text{,}\) give an antiderivative of \(f(x)+g(x)\text{.}\)
You should find that \(1/x\) has two types of antiderivatives, depending on whether \(x \gt 0\) or \(x\lt 0\text{.}\) In one expression, give a formula for \(\ds \int \frac{1}{x}\, dx\) that takes these different domains into account, and explain your answer.
