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

Sec. 5] In t egration of Rational Functions 123
Consequently,
dx 1
, , , , , ,
,
/= \ \ 7+ r- = ln JC In JC 1
\ ; 2 r-f C.
' ' ' '
A- 1
f dx f* dx , p
'
J X J jc1 J (*l)
If the polynomial Q (x) has complex roots a ib of multiplicity k, then
partial fractions of the form
will enter into the expansion (2). Here,
and A lt B lt .., A k , B k are undetermined coeflicients which are determined
by the methods given above For k~\, the fraction (5) is integrated directly;
for k>\, use is made of the reduction method; here, it is first advi-
sable to represent the quadratic trinomial x z + px~{-q in the form ( x-\-~ \
q ~] and make the substitution A--J- = z.
Example
3. Find
Solution. Since
then, putting x -\-2---z, wo got
A 2
-| 4x i
5-
r== r *\._ dz=z r _j_^ r Hit i!ini 2
jrc tan ? - - -
j2 ^
-- -- arc tan z=
2. The Ostrogradsky method. If Q (A) has multiple roots, then
P(x) A'(v) p r "- (.Y) A
(6)
where Q, (A:)
is the greatest common divisor of the polynomial Q (x) and its
derivative Q' (A-);
X (A-) and Y (x) are polynomials with undetermined coefficients, whose degrees
arc, respectively, less by unity than those
The undetermined coeflicients of the
of Q, (A-) and
polynomials
Q 2 (x).
X (x) and Y (x) are
computed by differentiating the 4. Find
identity (6).
Example
C
dx
} U'-
~
(5)
-f