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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.

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