]> 19.7 Partial Fraction Decomposition

## 19.7 Partial Fraction Decomposition

We know how to anti-differentiate a function of the form $( x − a ) b$ for any $a$ and $b$ . This will allow us to find the anti-derivative of a rational function $p ( x ) q ( x )$ if we can reduce it to a sum of terms of that form and possibly a polynomial instead.

If $p ( x )$ has a higher degree than $q ( x )$ we can extract a quotient polynomial $s ( x )$ by a process akin to long division called synthetic division.

We may then be left with a remainder polynomial $r ( x )$ . We know how to anti-differentiate $s ( x )$ so the task of anti-differentiating $p ( x ) q ( x )$ reduces to anti-differentiating $r ( x ) q ( x )$ where the numerator has lower degree than the denominator.

Suppose now that we can factor $q$ into factors like $( x − a )$ or $( x − b ) 3$ or $( ( x − d ) 2 + c 2 ) m$ .

The wonderful fact is that the expression $r ( x ) q ( x )$ can be separated into terms each of which has the form $( x − a ) − b$ or $( ( x − d ) 2 + c 2 ) − b$ or $x ( ( x − d ) 2 + c 2 ) b$ for some $a ' s$ , $d ' s$ and $c ' s$ and integer values of $b$ , each of which can be anti-differentiated.

And here is a procedure for separating it.

Suppose the denominator $q$ can be factored into $( x − b ) k t ( x )$ such that $t ( b )$ is not 0.

And suppose we find the first $k$ terms of the Taylor series expansion of $r ( x ) t ( x )$ about $x = b$

$r ( x ) t ( x ) = A + B ( x − b ) + C ( x − b ) 2 + ⋯ + J ( x − b ) k − 1 + ⋯$

Then the terms in $r ( x ) q ( x )$ that involve inverse powers of $( x − b )$ are given as follows:

If $k = 1$ there is only one term, $A ( x − b ) − 1$ , and $A$ is given by $r ( b ) t ( b )$ ;

For $k = 2$ we have $A ( x − b ) − 2 + B ( x − b )$ where $A$ is as before while $B$ is $( r ( x ) t ( x ) ) ' | x = b$ ; and so on.
There are similar rules for quadratic factors.

The process of separating the denominator $q$ in this manner is called "the method of partial fractions".

We review the various methods again in Section 27.1 and some integrals to practice on are given in Section 27.3 . We apologize for the redundancy.