Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
Example A1.18
Show that
Z 1
1
x 2 e ax2 dx ˆ 1=2
2a 3=2 :
Solution:
Let us ®rst consider the integral
b
I ˆ
Z 1
0
e ax2 dx
`Integration-by-parts' gives
Z c
c
I ˆ e ax2 dx ˆe ax2 x
‡ 2
from which we obtain
Z c
x 2 e ax2 dx ˆ 1
2a
b
Z c
b
b
Z c
b
ax 2 e ax2 dx;
c
e ax2 dx e ax2 x
:
We let limits b and c become 1 and ‡1, and thus obtain the desired result.
b
Problem A1.23Z
Evaluate I ˆ
xe x dx ( constant).
(3) Partial fractions: Any rational function P…x†=Q…x†, where P…x† and Q…x†
are polynomials, with the degree of P…x† less than that of Q…x†, can be written as
the sum of rational functions having the form A=…ax ‡ b† k ,
…Ax ‡ B†=…ax 2 ‡ bx ‡ c† k , where k ˆ 1; 2; 3; ... which can be integrated in
terms of elementary functions.
Example A1.19
3x 2
…4x 3†…2x ‡ 5† 3 ˆ
5x 2 x ‡ 2
…x 2 ‡ 2x ‡ 4† 2 …x 1† ˆ
A
4x 3 ‡ B
…2x ‡ 5† 3 ‡ C
…2x ‡ 5† 2 ‡ D
2x ‡ 5 ;
Ax ‡ B
…x 2 ‡ 2x ‡ 4† 2 ‡ Cx ‡ D
x 2 ‡ 2x ‡ 4 ‡ E
x 1 :
Solution: The coecients A, B, C etc., can be determined by clearing the fractions
and equating coecients of like powers of x on both sides of the equation.
Problem A1.24
Evaluate
Z
I ˆ
6 x
…x 3†…2x ‡ 5† dx:
532