Many operations in mathematics have inverses: addition...subtraction, multiplication...division, powers...roots. In each, the second operation undoes the first. Differentiation and Integration work in the same fashion. When you integrate a function you are, in effect, undoing a derivative.
The function
has the derivative
Reversing this process is called Anti-differentiation. Therefore:
There is one problem though, the function is also the derivative of the following functions:
It looks like the new function F(x) (as it is commonly notated) that is achieved when you reverse the process of differentiation, is not a unique one, but a whole set of functions. Therefore, the anti-derivative of 4x equals 2x^2+C where C represents some constant.
If the derivatives of two functions are equal, then the functions differ by, at most, a constant. What you are doing with this operation is taking the Indefinite Integral. The symbol used for this operation is the following
You will use the indefinite integral to represent all anti-derivatives
of f(x) and write:
The arbitrary constant c is called the constant of integration.
d is the Differential Operator which is Pre-defined in LiveMath and takes total derivatives.
And x in the above expression represents the independent variable of the function.
As was stated in the first paragraph of this section, when you integrate a function you are, in effect, undoing a derivative. The first example will demonstrate what is meant by this and will introduce the method used to Integrate in LiveMath.
You will use the term integration interchangeably with anti-differentiation (properly spelled...antidifferentiation) at this point. Keep in mind though, integration refers to both Definite and Indefinite integration. The indefinite integral represents all anti-derivatives of a particular function while the Definite integral may use anti-derivatives, or it may refer to a limit of a sum as you will see in the next section.
5.1 Undoing the DerivativeThere are two integration Ops in Livemath. Both are located on the palette and can be activated by either clicking on them or by using command key equivalents. The first is the Indefinite Integral, and can be activated by typing a $ sign (Shift-4). The second is the Definite Integral and is activated by typing Command-J.
LiveMath will not input the differential operator when you use Shift-4 ($) to activate the Indefinite Integral. This will allow you to Apply the Op to expressions that already have the differential operator. This is very useful when manipulating Differential Equations, as you will see later in the chapter.
or
The Differential Operator must be declared as the "Pre-defined" Differential Operator (a D-Linear Operator) in the declarations. This has been done for you in all New notebooks that are opened. If, however, you have overridden this by declaring d as a variable (or anything else), you will have problems with these type of problems. If you encounter problems, check your declarations!
In other words, if you Integrate a derivative you will get the original function back. When you are integrating, you are undoing a derivative!!
As you discovered in the study of derivatives, there are certain procedures, or rules, which allowed you to differentiate functions. Although the rules to integrate are not as inclusive, they are available, and the goal of the student, more so than with differentiation, is to choose the best one.
A very important point must now be made at this time. LiveMath, as with most math software, will have some limitations when it comes to integration. With differentiation, the rules discussed in the prior chapters will allow you to find the derivative of most continuous functions. In fact, LiveMath will do this automatically, as you have seen. This is not the case when you try to find anti-derivatives. Some functions just do not have an anti-derivative and others have no elementary formulas allowing you to integrate them.
Curious students have probably already tried to integrate some functions and most likely have been dismayed when the computer did not provide an answer for them. As you study the integration rules in this chapter, you may want to refer back to Chapter #3 to the analogous differentiation rule.
You found in Chapter #3 that the rule for taking derivatives of the power x^n is the following:
To find an Anti-derivative of x^n you reverse the procedure. You increase the exponent by 1 and divide by the new exponent, n + 1. You mathematically show this as:
Using LiveMath you can verify this rule by taking the derivative of this new function:
NOTE: Make sure that n is declared a constant and not a variable.
5.2 Power Rule for IntegralsUse the following function as your first example. The problem is split into several parts to show you how the constant of integration comes into play and to show how, with Auto Simplify off, you can use LiveMath to define the rule itself. Using Example #6.1 as a guide, enter the Integral into a new Prop. Switch Auto Simplify and Auto Casing Simplify on or off as indicated.
With Auto Casing turned on the full Integral, with the arbitrary constant, is generated. The constant has a numbered subscript so that if you are doing more than one problem the constants can be kept separate. Each newly generated constant will have a different, sequentially numbered, subscript.
To control the label and its subscript choose Arbitrary Constants... from Computation Prefs under the Calculate menu. The following dialog box will appear. This is where you can change the labeling conventions of the subscript.
This has been a rather involved example, the purpose of which is to show how LiveMath solves these integrals. As you work through these problems, the use of Auto Simplify and Auto Casing will help in the understanding of the underlying concepts of Integration.
The next example uses LiveMath again to show you how the Power Rule works.
5.3 Power Rule : Another Example
When you Integrate a function f(x), what you are doing is finding a formula that gives all the functions F(x) + c that could possibly have f(x) as a derivative. The functions that result are the anti-derivatives of f(x). The only difference between all of them is the Constant of Integration. When you graph several of these anti-derivatives in the same Graph Theory, you will see that these functions shift up or down on the y-axis.
5.4 What is the Constant of Integration??
You may ask at this point, "What good does it do to take the integral of a function if all you get is an infinite number of solutions, separated by this constant c ? " One of the more important things you can do with calculus is to recover a function from its rate of change (a derivative) and a single known "initial" value.
What is meant by this is the following:
By differentiating a function you are determining its rate of change. Because this original function has the constant attached to it (even if it is zero), the solution is unique, in other words, it has only one derivative. On the other hand, if you go backwards and are given a function that represents a derivative of another function (the rate of change), that function can have several solutions, or anti-derivatives.
If you are given a value (called an initial value), then you will be able to pick out of all the anti-derivatives the correct one! This is what is called an Initial Value Problem. If you are given the function that represents a rate of change, (2x) above, the equation dy/dx = 2x represents, what is called a Differential Equation. If you know an initial value, you will be able to find the correct anti-derivative out of the family of anti-derivatives for that function. You will be introduced to this concept later in the chapter.
