Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

From the last two equations it is clear that integration can be considered as

the inverse of differentiation. However, we see from the above analysis that the

lower limitais arbitrary and so differentiation does not have auniqueinverse.

Any functionF(x) obeying (2.28) is called anindefinite integraloff(x), though

any two such functions can differ by at most an arbitrary additive constant. Since

the lower limit is arbitrary, it is usual to write

F(x)=

∫x

f(u)du (2.29)

and explicitly include the arbitrary constant only when evaluatingF(x). The

evaluation is conventionally written in the form
∫
f(x)dx=F(x)+c (2.30)

wherecis called theconstant of integration. It will be noticed that, in the absence

of any integration limits, we use the same symbol for the arguments of bothf

andF. This can be confusing, but is sufficiently common practice that the reader

needs to become familiar with it.

We also note that the definite integral off(x) between the fixed limitsx=a

andx=bcan be written in terms ofF(x). From (2.27) we have
∫b

a

f(x)dx=

∫b

x 0

f(x)dx−

∫a

x 0

f(x)dx

=F(b)−F(a), (2.31)

wherex 0 isanythird fixed point. Using the notationF′(x)=dF/dx,wemay

rewrite (2.28) asF′(x)=f(x), and so express (2.31) as
∫b

a

F′(x)dx=F(b)−F(a)≡[F]ba.

In contrast to differentiation, where repeated applications of the product rule

and/or the chain rule will always give the required derivative, it is not always

possible to find the integral of an arbitrary function. Indeed, in most real phys-

ical problems exact integration cannot be performed and we have to revert to

numerical approximations. Despite this cautionary note, it is in fact possible to

integrate many simple functions and the following subsections introduce the most

common types. Many of the techniques will be familiar to the reader and so are

summarised by example.

2.2.3 Integration by inspection

The simplest method of integrating a function is by inspection. Some of the more

elementary functions have well-known integrals that should be remembered. The

reader will notice that these integrals are precisely the inverses of the derivatives