142 Chapter 5Integration
EXAMPLE 5.10
(i) Find the even and odd components off(x) 1 = 1 e
x, (ii) evaluate the definite integrals
off(x)and of its even and odd components over the rangex 1 = 1 − 1 tox 1 = 1 + 1.
(i) even:
odd:
These are the hyperbolic functionscosh 1 xandsinh 1 x(Equation (3.47) and Figure 3.22).
(ii)
Only the symmetric component makes a nonzero contribution to the integral off(x).
0 Exercises 40–47
The concept of the parity or symmetry of functions is widely used in the physical
sciences; the relevant branch of mathematics is called group theory. In molecular
chemistry, the point groups are used to describe, for example, the symmetry properties
of molecular wave functions and of the normal modes of vibration of molecules, and
are used to explain and predict the allowed transitions between energy levels that are
observed in molecular spectra. In solid-state chemistry, the space groups describe the
symmetry properties of lattices and the structure of X-ray diffraction spectra.
5.4 The integral calculus
Lety 1 = 1 f(x)be a function of x, continuous in the intervala 1 ≤ 1 x 1 ≤ 1 b, as in Figure 5.4.
We postulated in Section 5.3 that the ‘area under the curve’, the shaded region in
Figure 5.4, is given by the value of the definite integral off(x)from ato b. We now
look at how this result is derived.
To obtain an estimate of the area, we divide the interval ato binto nsubintervals by
choosingn 1 − 11 arbitrary points on the x-axis, with
a 1 = 1 x
01 < 1 x
11 < 1 x
21 <1-1< 1 x
n− 11 < 1 x
n1 = 1 b (5.28)
and divide the area into nstrips by vertical lines at these points as shown in Figure 5.11.
ZZ
−+−−+−−+=−
=+
1111111
2
f x dx e e dx e e
xx x()
−−
=
x0
ZZ
−++−+−−+=+
=−
1111111
2
f x dx e e dx e e
xx x()
−−−
=+
xee
1ZZ
−+−+−+−==
=+
1111111fxdx edx e e e
xx()
fx fx f x e e
xx−−=−−
=−
() () ( )
1
2
1
2
fx fx f x e e
xx+−=+−
=+
() () ( )
1
2
1
2