Integration by partial fractions

Error

There was an error loading the page.

Contributors

Bill Foster

Feedback

From users who are not members of Bill's workspace :

Christian Lawson-Perfect

said

Ready to use

9 years, 11 months ago

History

○

Christian Lawson-Perfect 9 years, 11 months ago

Gave some feedback: Ready to use

○

Bill Foster 55 years, 2 months ago

Created this.

Name	Status	Author	Last Modified
Integration by partial fractions	Ready to use	Bill Foster	01/06/2016 09:47
Simon's copy of Integration by partial fractions	draft	Simon Thomas	06/03/2019 15:05

There are 9 other versions that do you not have access to.

Question statement

Find the following integral.

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

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	7
c	integer	5
b	integer	-7
s3	integer	1
s2	integer	-1
s1	integer	1
b1	integer	-7
c1	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`

← 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.

Advice

Using partial fractions we have to find $A$ and $B$ such that:

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

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

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

We now identify coefficients on both sides of this equation.

Identifying coefficients:

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

Coefficent $x$ : $\simplify[std]{A + B = 0}$ which gives $A = -B$

Hence we obtain $\displaystyle \simplify[std]{A = {c}/{b-a}}$ and $\displaystyle \simplify[std]{B={-c}/{b-a}}$

Which gives:

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

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

Or equivalently: $\displaystyle I= \simplify[std]{({c}/{b-a})*(ln((x+{a})/(x+{b})))+C}$

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 used in the following exams:

Loading...

Error

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

Loading...

Error

Integration by partial fractions

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`