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7.2. INTEGRATION OF RATIONAL EXPRESSIONS 171
7.2 Integration of Rational Expressions
The integration of polynomials is trivial:
∫
n
∑
i=0
a i x i =
Since any rational expression f(x) ∈ K(x) can be written as
n∑
i=0
1
i + 1 a ix i+1 . (7.4)
f = p + q r with { p, q, r ∈ K[x]
deg(q) < deg(r)
, (7.5)
and p is always integrable by (7.4), we have proved the following (trivial) result:
we will see later that its generalisations, Lemmas 10 and 11, are not quite so
trivial.
Proposition 55 (Decomposition Lemma (rational expressions)) In the
notation of (7.5), f is integrable if, and only if, q/r is.
q/r with deg(q) < deg(r) is generally termed a proper rational function, but,
since we are concerned with the algebaric form of expressions in this chapter,
we will say “proper rational expression.”.
7.2.1 Integration of Proper Rational Expressions
In fact, the integration of proper rational expressions is conceptually trivial (we
may as well assume r is monic, absorbing any constant factor in q):
1. perform a square-free decomposition (Definition 32) of r = ∏ n
i=1 ri i ;
2. factorize each r i completely, as r i (x) = ∏ n i
j=1 (x − α i,j);
3. perform a partial fraction decomposition of q/r as
q
r = q
∏ n
i=1 ri i
=
n∑
q i
r i i=1 i
=
n∑ ∑n i
i∑
i=1 j=i k=1
β i,j,k
(x − α i,j ) k ; (7.6)
4. integrate this term-by-term, obtaining
∫ q
r =
∑
n ∑n i
i∑
i=1 j=i k=2
−β n i,j,k
(k − 1)(x − α i,j ) k−1 + ∑
∑n i
i=1 j=i
From a practical point of view, this approach has several snags:
β i,j,1 log(x − α i,j ).
(7.7)
1. we have to factor r, and even the best algorithms from the previous chapter
can be expensive;