INTEGRATION AS THE INVERSE OF DIFFERENTIATION - General Methods of Integration - The Calculus Primer

The Calculus Primer (2011)

Part XII. General Methods of Integration


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 = image;

y= xn, x= (y)1/n;

y = ax, x = loga 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 image to denote the process of integration. We write, for example,

image3x2dx = x3,

which is read: “the integral of 3x2 dx is x3.” The symbol image 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 image 10x dx = 5x2.

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

and image sin x dx = − cos x.

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

and image

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

and image 2 cos 2x dx = sin 2x.

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

image 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.