Page 45 - Physical Chemistry

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Chapter 1 Integral Calculus
Thermodynamics Frequently one wants to find a function y(x) whose derivative is known to be a certain
function f(x); dy/dx f(x). The most general function y that satisfies this equation is
called the indefinite integral (or antiderivative) of f(x) and is denoted by f(x) dx.

If dy>dx f 1x2 then y f 1x2 dx (1.52)*

The function f (x) being integrated in (1.52) is called the integrand.
Since the derivative of a constant is zero, the indefinite integral of any function
contains an arbitrary additive constant. For example, if f(x) x, its indefinite integral
1
2
y(x) is x C, where C is an arbitrary constant. This result is readily verified by
2
1
2
showing that y satisfies (1.52), that is, by showing that (d/dx) ( x C) x. To save
2
space, tables of indefinite integrals usually omit the arbitrary constant C.
From the derivatives given in Sec. 1.6, it follows that
af 1x2 dx a f 1x2 dx, 3 f 1x2 g1x24 dx f 1x2 dx g1x2 dx

(1.53)*
x n 1
dx x C, x dx C where n 1 (1.54)*
n
n 1
1 dx ln x C, e dx e a ax C (1.55)*
ax
x

cos ax sin ax
sin ax dx C, cos ax dx C (1.56)*
a a
where a and n are nonzero constants and C is an arbitrary constant. For more compli-
cated integrals than those in Eqs. (1.53) through (1.56), use a table of integrals or the
website integrals.wolfram.com, which does indefinite integrals at no charge.
A second important concept in integral calculus is the definite integral. Let f (x) be
a continuous function, and let a and b be any two values of x. The definite integral of
f between the limits a and b is denoted by the symbol

b
f1x2 dx (1.57)

a
The reason for the resemblance to the notation for an indefinite integral will become
clear shortly. The definite integral (1.57) is a number whose value is found from the
following definition. We divide the interval from a to b into n subintervals, each of
width x, where x (b a)/n (see Fig. 1.15). In each subinterval, we pick any point
we please, denoting the chosen points by x , x , . . . , x . We evaluate f(x) at each of
1 2 n
the n chosen points and form the sum
n
a f 1x 2¢x f 1x 2¢x f 1x 2¢x . . . f 1x 2¢x (1.58)
1
n
2
i
i 1
We now take the limit of the sum (1.58) as the number of subintervals n goes to in-
finity, and hence as the width x of each subinterval goes to zero. This limit is, by de-
finition, the definite integral (1.57):
b n
f 1x2 dx lim a f 1x 2¢x (1.59)
¢xS0 i
a i 1