Integration by partial fractions. (Video)

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Christian Lawson-Perfect

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Christian Lawson-Perfect 9 years, 11 months ago

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Bill Foster 55 years, 2 months ago

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Name	Status	Author	Last Modified
Integration by partial fractions. (Video)	Ready to use	Bill Foster	01/06/2016 09:47
Numbas demo: video	Ready to use	Christian Lawson-Perfect	25/08/2020 15:00
YJ's copy of Numbas demo: video	Ready to use	YJ Kim	01/06/2016 09:47
Simon's copy of Integration by partial fractions. (Video)	draft	Simon Thomas	06/03/2019 14:06
Simon's copy of Simon's copy of Integration by partial fractions. (Video)	draft	Simon Thomas	06/03/2019 14:09
Simon's copy of Integration by partial fractions. (Video)	draft	Simon Thomas	06/03/2019 15:18
Simon's copy of Simon's copy of Integration by partial fractions. (Video)	draft	Simon Thomas	06/03/2019 15:31
Simon's copy of Simon's copy of Simon's copy of Integration by partial fractions. (Video)	draft	Simon Thomas	06/03/2019 15:33

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Question statement

Find the following integral.

$I = \simplify[std]{Int(({c}*x+{d})/(x^2+{a+b}*x+{a*b}),x )}$

Input all numbers as fractions or integers and not decimals.

Input the constant of integration as $C$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	5
c	integer	9
b	integer	-8
d	integer	4
s3	integer	1
s2	integer	-1
s1	integer	1
b1	integer	-8
d1	integer	4

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xxxxxxxxxx
 
s1*random(1..9)

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Generated value: `integer`

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1. Gap Mathematical expression
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$I=\;$ Unnamed gap

Input all numbers as fractions or integers and not decimals.

Input the constant of integration as $C$ .

Click on Show steps for help if you need it. You will lose 1 mark if you do so.

There is a video in Show steps which solves a similar example.

Advice

First we factorise $\simplify[std]{x^2+{a+b}*x+{a*b}=(x+{a})*(x+{b})}$ . You can do this by spotting the factors or by completing the square.
Next we use partial fractions to find $A$ and $B$ such that:

$\displaystyle \simplify[std]{({c}*x+{d})/((x +{a})*(x+{b})) = A/(x+{a})+B/(x+{b})}$
Multiplying both sides of the equation by

$\displaystyle \simplify[std]{1/((x +{a})*(x+{b}))}$ we obtain:

$\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}*x+{d}} \Rightarrow \simplify[std]{(A+B)*x+{b}*A+{a}*B={c}*x+{d}}$

Identifying coefficients:

Constant term: $\simplify[std]{{b}*A+{a}*B = {d}}$

Coefficent $x$ : $\simplify[std]{A+B={c}}$ which gives $A =\var{c} -B$

On solving these equations we obtain $\displaystyle \simplify[std]{A = {d-a*c}/{b-a}}$ and $\displaystyle \simplify[std]{B={d-b*c}/{a-b}}$

Which gives:

$\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b})) = ({d-a*c}/{b-a})*(1/(x+{a}) )+({d-b*c}/{a-b})*(1/(x+{b}))}$

$\begin{eqnarray*} I &=& \simplify[std]{Int(({c}*x+{d})/(x^2+{a+b}*x+{a*b}),x )}\\ &=&\simplify[std]{Int(({c}*x+{d})/((x +{a})*(x+{b})),x )}\\ &=& \simplify[std]{({d-a*c}/{b-a})*(Int(1/(x+{a}),x)) +({d-b*c}/{a-b})Int(1/(x+{b}),x)}\\ &=& \simplify[std]{({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C} \end{eqnarray*}$

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This question is used in the following exams:

KR Calculus Exam by Kristina Raen in Kristina's workspace.
Video examples by Anthony Youd in Anthony's workspace.
Integration: Partial Fractions Practice by Bill Foster in Bill's workspace.
Bradford: Partial Fractions Practice by Bill Foster in Bill's workspace.
Maths Support: Integration using partial fractions by Christian Lawson-Perfect in Maths Support Wiki.
Random Exam by Ciar Mac an Bhaird in Ciarán's workspace.
Bath demo exam by martin reed in martin's workspace.
Logarthims by Aneela Arshad in Aneela's workspace.
Integration: 1 by Aneela Arshad in Aneela's workspace.
Calculous: 1 by Aneela Arshad in Aneela's workspace.
Integration by Bill Foster in Bill's workspace.
Nick's copy of Jack's copy of mathcentre: Integration using partial fractions by Nick Walker in Nick's workspace.
Integration using partial fractions by Simon Thomas in Partial fractions.
Integration using Partial Fractions by Kevin Bohan in Kevin's workspace.
Foundation Maths B - Week 8 Integration of partial fraction by Jinhua Mathias in Core Foundation Maths.

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Christian Lawson-Perfect 9 years, 11 months ago

Bill Foster 55 years, 2 months ago

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Integration by partial fractions. (Video)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 9 years, 11 months ago

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`