Indefinite integral by substitution

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Julie Crowley

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Julie Crowley 8 years, 5 months ago

Published this.

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Julie Crowley 11 years, 2 months ago

Created this as a copy of Indefinite integral by substitution.

Name	Status	Author	Last Modified
Indefinite integral by substitution	draft	Bill Foster	06/08/2020 08:41
Indefinite integral by substitution	draft	Julie Crowley	08/10/2016 21:58

Question statement

Find the following integral.

Input the constant of integration as $C$ .

Input all numbers as integers or fractions not as decimals.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
s1	integer	1
b	integer	3

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Data type

Value

xxxxxxxxxx
 
2*s1*random(1..5)

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Generated value: `integer`

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$I=\int\frac{\simplify[std]{{a}*x+{b}}}{(1-x^2)^{1/2}} \;dx$

$I=\;$ Unnamed gap

Input the constant of integration as $C$ .

Input all numbers as integers or fractions not as decimals.

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

Advice

Split the integral into two parts

$I=\int\frac{\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \;dx+\int\frac{\simplify[std]{{b}}}{(1-x^2)^{1/2}} \;dx$
For the integral

$I_1=\int\frac{\simplify[std]{{a}*x}}{(1-x^2)^{1/2}} \;dx$ use the substitution

$u=1-x^2$ and then

$du=-2xdx$ and we get

$\begin{eqnarray*}I_1&=&\simplify[std]{{-a}/2*Int((1 / (u^(1/2))),u)}\\\\ &=&\simplify[std]{({-a}/2)*(2u^(1/2))+C}\\ &=&\simplify[std]{({-a})*(1-x^2)^(1/2)+C} \end{eqnarray*}$
The other integral is a standard result:

$I_2=\simplify[std]{Int((({b}) / (1-x^2)^(1/2)),x)={b}*arcsin(x)+C}$
Putting these together gives:

$I=I_1+I_2=\simplify[std]{-{a}*(1-x^2)^(1/2)+{b}*arcsin(x)+C}$

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This question is used in the following exams:

Bridging Calculus Review by Julie Crowley in Waste.
All of Integration by Julie Crowley in Julie's workspace.
Bridging Calculus Review by Marlon Arcila in Tst.
Methods of Integration by Deirdre Casey in Deirdre's workspace.
Week 2 Day 4 by Julie Crowley in Julie's workspace.
Integration revision for second year by John Steele in John's workspace.
Methods of integration by Simon Thomas in Maths support.
Markus's copy of Methods of Integration by Markus Schmied in Markus's workspace.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Indefinite integral by substitution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Julie Crowley 8 years, 5 months ago

Julie Crowley 11 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`