Integration by Partial Fractions

A rational function is a ratio $\dfrac{P(x)}{Q(x)}$ of two polynomials. When $Q(x)$ factors

into simpler pieces, the rational function can be rewritten as a SUM of simpler fractions --

its partial fraction decomposition -- each of which is directly integrable using the

standard log/arctan forms of Section 5.

Proper vs improper. The decomposition method below applies directly only to a proper

rational function, where $\deg P<\deg Q$. If $P/Q$ is improper ($\deg P\ge\deg Q$), divide

$P(x)$ by $Q(x)$ FIRST by long division, writing $\dfrac{P(x)}{Q(x)}=T(x)+\dfrac{P_1(x)}{Q(x)}$,

where $T(x)$ is a polynomial (integrated term by term) and $P_1(x)/Q(x)$ is now proper.

Case: distinct linear factors. If $Q(x)=(x-a_1)(x-a_2)\cdots(x-a_n)$ with all $a_i$ distinct,

where each constant $A_i$ is found by clearing denominators (multiplying both sides by $Q(x)$)

and either comparing coefficients or substituting $x=a_i$, which makes every term except the

$A_i$ one vanish.

Case: repeated linear factor. A factor $(x-a)^k$ contributes $k$ terms,

$\dfrac{A_1}{x-a}+\dfrac{A_2}{(x-a)^2}+\cdots+\dfrac{A_k}{(x-a)^k}$, rather than a single term.

Case: irreducible quadratic factor. A quadratic factor $ax^2+bx+c$ with no real roots

(negative discriminant) contributes a term $\dfrac{Bx+D}{ax^2+bx+c}$ with a LINEAR numerator,

integrated using the numerator-splitting technique of Section 5 (writing $Bx+D$ as a multiple of

the denominator's derivative plus a constant remainder).

Worked illustration. For $\displaystyle\int\frac{3x+1}{(x-1)(x+2)}\,dx$: write

$\dfrac{3x+1}{(x-1)(x+2)}=\dfrac{A}{x-1}+\dfrac{B}{x+2}$, so $3x+1=A(x+2)+B(x-1)$. Setting $x=1$:

$4=3A\Rightarrow A=4/3$. Setting $x=-2$: $-5=-3B\Rightarrow B=5/3$. Hence

$\int\frac{3x+1}{(x-1)(x+2)}\,dx = \frac{4}{3}\ln|x-1|+\frac{5}{3}\ln|x+2|+C,$ …