## 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 = x*^{3}*,* *x* = ;

*y= x ^{n}, 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:

3*x*^{2} is the derivative of what function?

or, in differential notation:

3*x*^{2} *dx* is the differential of what expression?

The answer, of course, can be seen by inspection; the anti-derivative of 3*x*^{2} is *x*^{3}, which may be checked at once by differentiating *x*^{3}*.*

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,

3*x*^{2}*dx = x*^{3},

which is read: *“the integral of* 3*x*^{2} *dx is x*^{3}*.”* The symbol is called the *integral sign;* the expression to be integrated (3*x*^{2} 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*) = 5*x*^{2}, then *f′*(*x*) *dx* = 10*x dx,*

and 10*x dx =* 5*x*^{2}*.*

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 2*x*, then *f′* (*x*) *dx* = 2 cos 2*x* d*x*,

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

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:

*x*^{4} + 10, and *x*^{4} − 3.

The derivative in each case is 4*x*^{3}. Now, if we were given the expression 4*x*^{3}, and were asked to find its integral, it is obvious that we could not tell which constant to assign to the *x*^{4}; in fact, there are an infinite number of functions having the derivative 4*x*^{3}, differing only in the constant term. Hence we may write

4*x*^{3} *dx = x*^{4} *+ 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.*