Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION

Combining (2.23) and (2.24) withcset equal toashows that

∫b

a

f(x)dx=−

∫a

b

f(x)dx. (2.26)

2.2.2 Integration as the inverse of differentiation

The definite integral has been defined as the area under a curve between two

fixed limits. Let us now consider the integral

F(x)=

∫x

a

f(u)du (2.27)

in which the lower limitaremains fixed but the upper limitxis now variable. It

will be noticed that this is essentially a restatement of (2.21), but that the variable

xin the integrand has been replaced by a new variableu. It is conventional to

rename thedummy variablein the integrand in this way in order that the same

variable does not appear in both the integrand and the integration limits.

It is apparent from (2.27) thatF(x) is a continuous function ofx, but at first

glance the definition of an integral as the area under a curve does not connect with

our assertion that integration is the inverse process to differentiation. However,

by considering the integral (2.27) and using the elementary property (2.24), we

obtain

F(x+∆x)=

∫x+∆x

a

f(u)du

=

∫x

a

f(u)du+

∫x+∆x

x

f(u)du

=F(x)+

∫x+∆x

x

f(u)du.

Rearranging and dividing through by ∆xyields

F(x+∆x)−F(x)

∆x

=

1

∆x

∫x+∆x

x

f(u)du.

Letting ∆x→0 and using (2.1) we find that the LHS becomesdF/dx,whereas

the RHS becomesf(x). The latter conclusion follows because when ∆xis small

the value of the integral on the RHS is approximatelyf(x)∆x, and in the limit

∆x→0 no approximation is involved. Thus

dF(x)

dx

=f(x), (2.28)

or, substituting forF(x) from (2.27),

d

dx

[∫x

a

f(u)du

]

=f(x).