Partial Fractions Reference Sheet
The$method$of$partial'fraction'decomposition'(PFD)$allows$us$to$antidifferentiate$rational$expressions$
by$rewriting$them$in$terms$of$simpler$rational$expressions$that$we$know$how$to$antidifferentiate.$$
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Recall$from$algebra:$$Every'polynomial'with'real'coefficients'can'be'factored'into'linear'and'
irreducible'quadratic'factors.'$
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A$typical$problem$might$look$like$this:$
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!
2๐ฅ$โ 4๐ฅโ8
(๐ฅ)โ ๐ฅ)(๐ฅ)+4)๐๐ฅ
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Partial'Fraction'Decomposition'โ'The'Method'that'Always'Works'
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1. If$the$degree$of$the$polynomial$in$the$numerator$is$greater$than$or$equal$to$the$degree$of$the$
polynomial$in$the$denominator,$use'long'division'or'synthetic'division$to$rewrite$the$
integrand.$
2. Then$factor$the$denominator$into$linear$factors$and$irreducible$quadratic$factors.$
3. Repeated'Linear'Factors:$For$each$factor$of$the$form$(๐๐ฅ+๐)/,$the$partial$fraction$
decomposition$must$include$the$following$terms:$
$๐ด
๐๐ฅ+ ๐,๐ต
(๐๐ฅ+ ๐)),๐ถ
(๐๐ฅ+ ๐)$,โฆ, ๐
(๐๐ฅ+ ๐)/$
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4. Repeated'Quadratic'Factors:$For$each$factor$of$the$form$(๐๐ฅ)+๐๐ฅ +๐)/,$where$๐)โ4๐๐ < 0,$
the$partial$fraction$decomposition$must$include$the$following$terms:$
$๐ด๐ฅ+ ๐ต
๐๐ฅ)+๐๐ฅ+ ๐,๐ถ๐ฅ +๐ท
(๐๐ฅ)+๐๐ฅ+ ๐)),๐ธ๐ฅ +๐น
(๐๐ฅ)+๐๐ฅ +๐)$,โฆ, ๐บ๐ฅ + ๐ป
(๐๐ฅ)+๐๐ฅ+ ๐)/$
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5. Then$add$the$terms$from$steps$(3)$and$(4)$together.$This$is$form$of$the$partial$fraction$
decomposition$of$the$original$rational$expression.$
6. Multiply$the$form$from$(5)$by$the$LCD,$and$collect$like$terms$in$๐ฅ/,๐ฅ/?@,โฆ,๐ฅ),๐ฅ,$and$so$on.$$
7. Then$equate$the$coefficients$of$like$terms$on$both$sides.$
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We$can$do$this$because$two'polynomials'are'equal'if'and'only'if'their'coefficients'are'equal.''
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8. Solve$the$resulting$system$of$equations$for$the$constants$๐ด,๐ต,๐ถ,etc.$from$the$form$in$(5).$
Substitute$these$constants$into$that$form.$
9. Now$the$rational$expression$may$be$antidifferentiated$using$basic$rules,$yielding$expressions$
whose$antiderivatives$are$natural$logarithmic$functions,$power$functions$requiring$๐ข-
substitution,$and$often$arctangent$functions.$$
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There$are$short-cuts$for$linear$factors,$as$described$in$the$detailed$lesson$notes.$