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46: Applications of Partial Fractions © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

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Applications of Partial Fractions There are 2 topics where partial fractions are useful. finding some binomial expansions differentiating some algebraic fractions integrating some algebraic fractions

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Applications of Partial Fractions Algebraic fractions can be awkward to differentiate and changing them into partial fractions makes them much easier. e.g. Differentiate Solution: The partial fractions are We need the chain rule here.

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Applications of Partial Fractions Uses in Integration Problems Expressions Which Give Ln`s If the Top is Gradient of BottomAns = ln(bottom)

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Applications of Partial Fractions Top is NOT Gradient of Bottom so FIX it. The gradient of the bottom is 4 so put a 4 on the top line and divide by 2 so that the question has not changed The gradient of the bottom is 15x 2 so put a 15x 2 on the top line and by 4 / 15 so that the question has not changed

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Applications of Partial Fractions This constant... Simplifying the answer If the Top is Gradient of BottomAns = ln(bottom)

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Applications of Partial Fractions This constant... is usually replaced by, so we get which can be tidied up using the 1 st law of logs to give Simplifying the answer

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Applications of Partial Fractions We’ll take (b) as our 1 st example using partial fractions. e.g.1 Find Solution: Factorise the bottom and express as 2 partial fractions so,

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Applications of Partial Fractions Both terms give us log integrals. It’s safer to separate the terms and adjust the constants for each. ( 2)

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Applications of Partial Fractions The 3 logs can now be simplified: using the 3 rd law: using the 1 st and 2 nd laws:

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Applications of Partial Fractions Solution: e.g. 2 Write as the sum of 3 partial fractions and then find where

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Applications of Partial Fractions The partial fractions are: The 2 nd and 3 rd integrals give logs but the 1 st doesn’t. The 1 st is of the form so, (This integral comes up often so it’s worth remembering)

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Applications of Partial Fractions so, and Since one term doesn’t give a log, we don’t gain by using instead of C.

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Applications of Partial Fractions e.g.3 Express as a single log Solution: Multiply by : So:

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Applications of Partial Fractions It makes the integration easier if we now write

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Applications of Partial Fractions Rational functions can be integrated if SUMMARY they are of the form or if they can be written as partial fractions with a linear denominator the fraction will integrate to a log with a repeated factor the fraction will integrate to another fraction e.g. If all terms integrate to logs, replaces C

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Applications of Partial Fractions Exercises 1.Express the following as a single logarithm (a) 2. Find (b)

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Applications of Partial Fractions 1(a) Solutions:

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Applications of Partial Fractions (b) The partial fractions are so,

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Applications of Partial Fractions 2.

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Applications of Partial Fractions

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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Applications of Partial Fractions There are 2 topics where partial fractions are useful. finding some binomial expansions integrating some algebraic fractions differentiating some algebraic fractions

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Applications of Partial Fractions SUMMARY To find a binomial expansion for an algebraic fraction with a denominator that factorises into linear factors: Find the partial fractions. For each partial fraction, write the denominator with a negative power. Expand each part and write the values for which the series converges. Find the validity of the entire expansion by choosing the most stringent of the restrictions on x. Combine the expansions. For expressions of the form, remove from the brackets.

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Applications of Partial Fractions We expand the 2 fractions separately. We have: Replace n by e.g.

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Applications of Partial Fractions The 2 nd fraction is For the binomial we must have, so we take outside the brackets: so we can save time by replacing x by in that. The 1 st fraction gave so,

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Applications of Partial Fractions So,

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Applications of Partial Fractions Algebraic fractions can be awkward to differentiate and changing them into partial fractions makes them much easier. e.g. Differentiate Solution: The partial fractions are We need the chain rule here.

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Applications of Partial Fractions Rational functions can be integrated if SUMMARY they are of the form or if they can be written as partial fractions fractions with a linear denominator will integrate to logs fractions with a repeated factor will integrate to another fraction e.g. If all terms integrate to logs, replaces C

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Applications of Partial Fractions and There’s an important difference. (a) is of the form and can be integrated directly. (b) is not of this form and cannot be converted to it by using a multiplying constant. Instead, we use partial fractions. Consider

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Applications of Partial Fractions There is one thing it’s useful to notice about (a) before we move to (b) This constant... is usually replaced by, so we get which can be tidied up using the 1 st law of logs to give

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Applications of Partial Fractions e.g.1 Find Solution: We found earlier that so, Both terms give us log integrals. It’s safer to separate the terms and adjust the constants for each.

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Applications of Partial Fractions ( 2) The 3 logs can now be simplified: using the 3 rd law: using the 1 st and 2 nd laws:

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Applications of Partial Fractions e.g.2 Find The 2 nd and 3 rd integrals give logs but the 1 st doesn’t. The 1 st is of the form Solution: (This integral comes up often so it’s worth remembering)

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Applications of Partial Fractionsso, and Since one term doesn’t give a log, we don’t gain by using instead of C.

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