1—Basic Stuff 41.2 Parametric Differentiation
The integration techniques that appear in introductory calculus courses include a variety of methods of
varying usefulness. There’s one however that is for some reason not commonly done in calculus courses:
parametric differentiation.It’s best introduced by an example.
∫∞0xne−xdx
You could integrate by partsntimes and that will work. For example,n= 2:
=−x^2 e−x
∣∣
∣
∣
∞0+
∫∞
02 xe−xdx= 0− 2 xe−x
∣∣
∣
∣
∞0+
∫∞
02 e−xdx= 0− 2 e−x
∣∣
∣
∣
∞0= 2
Instead of this method, do something completely different. Consider the integral
∫∞0e−αxdx (1.5)
It has the parameterαin it.The reason for this will be clear in a few lines. It is easy to evaluate, and is
∫∞
0e−αxdx=
1
−α
e−αx
∣∣
∣∣
∞0=
1
α
Now differentiate this integral with respect toα,
d
dα
∫∞
0e−αxdx=
d
dα
1
α
or −∫∞
0xe−αxdx=
− 1
α^2
And again and again: +
∫∞
0x^2 e−αxdx=
+
α^3
, −
∫∞
0x^3 e−αxdx=
− 2. 3
α^4
Thenthderivative is
±
∫∞
0xne−αxdx=
±n!
αn+
(1.6)
Setα= 1and you see that the original integral isn!. This result is compatible with the standard
definition for0!. From the equationn! =n.(n−1)!, you take the casen= 1, and it requires0! = 1
in 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. Also see problem1.47for
a variation on this theme.
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 problem1.42.
1.3 Gaussian Integrals
Gaussian integrals are an important class of integrals that show up in kinetic theory, statistical mechan-
ics, quantum mechanics, and any other place with a remotely statistical aspect.
∫dxxne−αx
2The simplest and most common case is the definite integral from−∞to+∞or maybe from 0 to∞.