Simon's copy of Integration by partial fractions. (Video)

Error

There was an error loading the page.

Contributors

History

○

Simon Thomas 6 years ago

Created this as a copy of Integration by partial fractions. (Video).

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

There are 3 other versions that do you not have access to.

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
d	integer	-2
d1	integer	-2
s2	integer	-1
s3	integer	-1
b1	integer	-2
s1	integer	-1
b	integer	-2
c	integer	5

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
s1*random(1..9)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

-5

← Depends on:

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Mathematical expression
  Add
1. Step Information only

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$I=\;$ 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

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

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

One way to find A and B is to compare coefficients:

Identifying coefficients:

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

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

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

Which gives:

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

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

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Information only

This question is not used in any exams.

Loading...

Error

Simon's copy of Integration by partial fractions. (Video)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Simon's copy of Integration by partial fractions. (Video)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`