Mathematical Methods for Physicists: A concise introduction - Site Map

SOME USEFUL METHODS OF INTEGRATION
Some useful methods of integration
(1) Changing variables: We use a simple example to illustrate this common
procedure. Consider the integral
I ˆ
Z 1
0
e ax2 dx;
which is equal to …=a† 1=2 =2: To show this let us write
Then
I 2 ˆ
Z 1
0
I ˆ
e ax2 dx
Z 1
0
Z 1
0
e ax2 dx ˆ
e ay2 dy ˆ
Z 1
0
e ay2 dy:
Z 1 Z 1
0
0
e a…x2 ‡y 2† dxdy:
We now rewrite the integral in plane polar coordinates …r;†:
x 2 ‡ y 2 ˆ r 2 ; dxdy ˆ rdrd . Then
Z 1 Z =2
I 2 ˆ e ar2 rddr ˆ Z 1
e ar2 rdr ˆ
1
ear2
ˆ
2
2 2a 4a
and
0
0
I ˆ
Z 1
(2) Integration by parts: Since
0
0
e ax2 dx ˆ…=a† 1=2 =2:
d
dx … uv†
ˆ u dv
dx ‡ v du
dx ;
where u ˆ f …x† and v ˆ g…x†, it follows that
Z
u
dv Z
dx ˆ uv v
du
dx:
dx
dx
This can be a useful formula in evaluating integrals.
0
Example A1.17
Evaluate I ˆ R tan 1 xdx
Solution: Since tan 1 x can be easily di€erentiated, we write
I ˆ R tan 1 xdx ˆ R 1 tan 1 xdx and let u ˆ tan 1 x; dv=dx ˆ 1. Then
Z
I ˆ x tan 1 xdx
x
1 ‡ x 2 ˆ x tan1 x 1 2 log…1 ‡ x2 †‡c:
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