The Calculus Primer (2011)

Part XII. General Methods of Integration

Chapter 44. INTEGRATION AS THE INVERSE OF DIFFERENTIATION

12—1. The Integral as an Anfi-derivative. The reader is already familiar with inverse operations and inverse functions. Consider the following:

y = x³, x = ;

y= xⁿ, x= (y)^1/n;

y = a^x, x = log_a y;

y = tan θ, θ = arc tan y.

Here the right-hand members in either column are the inverses of the right-hand members of the other column, and are obtained in each case by inverse operations. Similarly, the inverse operation of finding a derivative is known as integration.

In the Differential Calculus we answered the question: given a function, what is its derivative? Now we shall concern ourselves, in the Integral Calculus, with the inverse problem: given a derivative, what is the function from which it was obtained? For example, integral calculus is concerned with questions such as:

3x² is the derivative of what function?

or, in differential notation:

3x² dx is the differential of what expression?

The answer, of course, can be seen by inspection; the anti-derivative of 3x² is x³, which may be checked at once by differentiating x³.

The process of finding the function, given its derivative, is called integration; the result is called the integral. Hence an integral may be regarded as an anti-derivative, or as an anti-differential. The operations of the Differential Calculus may be symbolized as follows:

or, in differential notation,

df(x) = f′(x)dx.

In the Integral Calculus we shall solve problems such as:

1. Find a function f(x) whose derivative f′ (x) = ø (x) is given;

2. Given the differential of a function, find the function itself; i.e., df(x) = f′ (x) dx = ø(x) dx.

The function, f(x), which we are seeking, is the integral of the given differential; we use the symbol to denote the process of integration. We write, for example,

3x²dx = x³,

which is read: “the integral of 3x² dx is x³.” The symbol is called the integral sign; the expression to be integrated (3x² in our illustration) is called the integrand. To illustrate integration from this point of view, namely, as the process of finding an anti-derivative, consider the following examples.

EXAMPLE 1. If f(x) = 5x², then f′(x) dx = 10x dx,

and 10x dx = 5x².

EXAMPLE 2. If f(x) = − cos x, then f′(x) dx = sin x dx,

and sin x dx = − cos x.

EXAMPLE 3. If f(x) = log x, then f′ (x) dx = dx,

and

EXAMPLE 4. If f(x) = sin 2x, then f′ (x) dx = 2 cos 2x dx,

and 2 cos 2x dx = sin 2x.

It should therefore be clear that the operation represented by the symbol is the inverse of the operation denoted by the symbol … dx. If we are using differential notation, then d and are symbols of inverse operations.

12—2. The Constant of Integration. Suppose we were to find the derivative of each of the following expressions:

x⁴ + 10, and x⁴ − 3.

The derivative in each case is 4x³. Now, if we were given the expression 4x³, and were asked to find its integral, it is obvious that we could not tell which constant to assign to the x⁴; in fact, there are an infinite number of functions having the derivative 4x³, differing only in the constant term. Hence we may write

4x³ dx = x⁴ + C,

where the symbol C represents an arbitrary constant and is called the constant of integration. Thus the constant C may have any one of an infinite number of values, and the number of functions obtained from a given integration is infinite, unless some additional condition of the problem permits the determination of the value of C. If we can find a definite value for C, then we refer to the result of the integration as a particular integral; if not, we speak of it as a general integral, or an indefinite integral We shall discuss the constant of integration again in a later chapter. For the present, we may summarize the concept involved by stating, without formal proof, the two following general principles:

I. If two functions differ by a constant, they have the same derivative.

II. If two functions have the same derivative, their difference is a constant.