Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 5] Integration of Rational Functions 121
i o K n i j - < < c dx
r\ I
\o\J. ---
\
v 2 dy
*~
I V~2 {-3* 2* 2
'
1270.
1 O *7 1 t
1272.
r d* 1273. J /"*-
J ' X X
1 1-7/1 f l/"o
,H4.
Jf~=. V % xdx
1265. 1
V>1J 5
^
1266. T - x ."~ - dx.
1277
1267. [ -,-^ dK. 197 n
J /5, 2 -2, f-1
1278
1268. 1'-^=. 2
, 279
-v
JYKI
Sec. 5. Integration of Rational Functions
J
f*
t'
g x cJJt
sin x dA
(
J T^PTTT^TlT-
Injfdi
i'
J x m
^ _ 2
1 ilnA._ !n K
t. The method of undetermined coefficients. Integration of a rational
function, after taking out the whole part, reduces to integration of the proper
rational fraction
where P (x) and Q (A-) are integral polynomials, and the degree of the numerator
P (x) is lower than that of the denominator Q (A-).
If
Q(jr) = (* a)*. . .(A'-/)\
where a, . . ., / are real distinct roots of the polynomial Q (x), and a, ....
K are natural numbers (root multiplicities), then decomposition of (1) into
partial fractions is justified:
^
To calculate the undetermined coefficients A lt A 2t ..., both sides of the
identity (2) are reduced to an integral form, and then the coefficients of
like powers of the variable x are equated (llrst method). These coefficients
may likewise be determined by putting [in equation (2) or in an equivalent
equation] x equal to suitably chosen numbers (second method).