Find indefinite integrals of hyperbolic functions

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Bill Foster	said	Ready to use	6 years, 5 months ago
Gabriele De Chiara	said	Has some problems	8 years, 11 months ago

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Bill Foster 6 years, 5 months ago

Gave some feedback: Ready to use

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Gabriele De Chiara commented 8 years, 11 months ago

Hi, I understand that the constant C should be inserted, however if F(x) is a primitive of f(x), both F(x)+C and F(x)+1+C are two possible solutions. I am looking for a way to test the F'(x)=f(x) or that the difference of the student's solution from mine is constant.

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Gabriele De Chiara 8 years, 11 months ago

Gave some feedback: Has some problems

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Created this.

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Question statement

Integrate the following functions $f(x)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
b	integer	-5
s2	integer	-1
s1	integer	-1
a1	integer	4
b1	integer	-1

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Name

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Value

xxxxxxxxxx
 
random(2..9)

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Description

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Mathematical expression
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Gap-fill

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

$f(x)=\simplify[std]{ cosh({a}x+{b})}$

$\displaystyle{\int f(x)\;dx=\;\;}$ Unnamed gap

You must include the constant of integration as $C$ .

Input all numbers as integers or fractions – not as decimals.

Advice

a) Since $\int \cosh(x)=\sinh(x)+C$ it follows that:

$\int \simplify[std]{cosh({a}x+{b})}\;dx = \frac{1}{\var{a}}\simplify[std]{ sinh({a}x+{b})}+C$

b) We perform integration by parts:

Using $\int u dv = uv - \int v du$ where:

$\begin{eqnarray*} &u&=x \Rightarrow du = dx \\ &dv& = \simplify[std]{sinh({a1}x+{b1})} \Rightarrow v= \simplify[std]{((cosh({a1}x+{b1}))/{a1})}\end{eqnarray*}$

Hence we have:

$\begin{eqnarray*} \int \simplify[std]{x*sinh({a1}x+{b1})}\;dx&=&\frac{1}{\var{a1}}\simplify[std]{x*cosh({a1}x+{b1})}-\frac{1}{\var{a1}} \int \simplify[std]{cosh({a1}x+{b1})}\;dx\\ &=&\frac{1}{\var{a1}}\simplify[std]{x*cosh({a1}x+{b1})}-\frac{1}{\var{a1^2}}\simplify[std]{sinh({a1}x+{b1})}+C \end{eqnarray*}$

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This question is used in the following exams:

Integration by parts by Newcastle University Mathematics and Statistics in Content created by Newcastle University.
Integration by parts by Henrik Skov Midtiby in SDS.
Luis's copy of Integration by parts by Luis Hernandez in MAT333.
Ann's copy of Integration by parts by Ann Smith in Calculus.
Integration by parts by Ann Smith in Calculus.
Integration by parts -section 8E by Kuldeep Singh in Kuldeep's workspace.
claire's copy of Integration by parts -section 8E by claire lines in claire's workspace.
Calculus by Morgiane Richard in Mathematics for Geosciences.

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Find indefinite integrals of hyperbolic functions

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Bill Foster 6 years, 5 months ago

Gabriele De Chiara commented 8 years, 11 months ago

Gabriele De Chiara 8 years, 11 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Find indefinite integrals of hyperbolic functions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 5 months ago

Gabriele De Chiara commented 8 years, 11 months ago

Gabriele De Chiara 8 years, 11 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`