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Home | 18.013A | Chapter 19 |
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We know how to anti-differentiate a function of the form for any and . This will allow us to find the anti-derivative of a rational function if we can reduce it to a sum of terms of that form and possibly a polynomial instead.
If has a higher degree than we can extract a quotient polynomial by a process akin to long division called synthetic division.
We may then be left with a remainder polynomial . We know how to anti-differentiate so the task of anti-differentiating reduces to anti-differentiating where the numerator has lower degree than the denominator.
Suppose now that we can factor into factors like or or .
The wonderful fact is that the expression can be separated into terms each of which has the form or or for some , and and integer values of , each of which can be anti-differentiated.
And here is a procedure for separating it.
Suppose the denominator can be factored into such that is not 0.
And suppose we find the first terms of the Taylor series expansion of about
Then the terms in that involve inverse powers of are given as follows:
If there is only one term, , and is given by ;
For
we have
where
is as before while
is
; and so on.
There are similar rules for quadratic factors.
The process of separating the denominator 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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