Mathematical Tools for Physics

1—Basic Stuff 6

Now differentiate this integral with respect toα,

d

dα

∫∞

0

e−αxdx=

d

dα

1

α

or −

∫∞

0

xe−αxdx=

− 1

α^2

And differentiate again and again:

+

∫∞

0

x^2 e−αxdx=

+

α^3

, −

∫∞

0

x^3 e−αxdx=

− 2. 3

α^4

Thenthderivative is

±

∫∞

0

xne−αxdx=

±n!

αn+

(6)

Setα= 1and you see that the original integral isn!. This result is compatible with the standard definition for
0!. From the equationn! =n.(n−1)!, you take the casen= 1. This requires0! = 1in order to make any
sense. This integral gives the same answer forn= 0.
The idea of this method is to change the original problem into another by introducing a parameter. Then
differentiate with respect to that parameter in order to recover the problem that you really want to solve. With
a little practice you’ll find this easier than partial integration.
Notice that I did this using definite integrals. If you try to use it for an integral without limits you can
sometimes get into trouble. See for example problem 42.

1.3 Gaussian Integrals
Gaussian integrals are an important class of integrals that show up in kinetic theory, statistical mechanics, quantum
mechanics, and any other place with a remotely statistical aspect.
∫
dxxne−αx

2

The simplest and most common case is the definite integral from−∞to+∞or maybe from 0 to∞.
Ifnis a positive odd integer, these are elementary,
∫∞

−∞

dxxne−αx

2

= 0 (nodd) (7)