Indefinite integral by substitution

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Name	Status	Author	Last Modified
Indefinite integral by substitution	draft	Bill Foster	06/08/2020 08:41
Julie's copy of Indefinite integral by substitution	draft	Julie Crowley	06/02/2017 15:34
Indefinite integral by substitution 2 (custom feedback)	draft	Clodagh Carroll	29/01/2020 16:43

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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	9
m	integer	5

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xxxxxxxxxx
 
random(1..9)

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Description

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

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Generated value: `integer`

→ Used by:

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"Unnamed part" - prompt
"Unnamed part" → "Unnamed gap" - Correct answer
"Unnamed part" → "Unnamed step" - prompt

Parts

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

Gap-fill

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$I=\simplify[std]{Int( sin(x)*({a} + {b}*cos(x))^{m},x)}$

Input all numbers as integers or fractions.

$I=\;$ Unnamed gap

Input the constant of integration as $C$ .

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

Advice

This exercise is best solved by using substitution.
Note that if we let $u=\simplify[std]{{a}+{b}cos(x)}$ then $du=\simplify[std]{({-b}*sin(x))*dx }$
Hence we can replace $\sin(x)\;dx$ by $\frac{1}{\var{-b}}\;du$ .

Hence the integral becomes:

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

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Taxonomy: mathcentre

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Indefinite integral by substitution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`