Simon's copy of Antiderivatives

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Simon Thomas 5 years, 11 months ago

Created this as a copy of Antiderivatives.

Name	Status	Author	Last Modified
Antiderivatives	draft	TEAME UCC	06/02/2017 13:30
Simple Antiderivatives	draft	Kieran Mulchrone	08/02/2017 14:35
cormac's copy of Antiderivatives	draft	cormac breen	23/02/2017 18:12
MA1002 Antiderivatives	Ready to use	Kieran Mulchrone	13/03/2019 07:44
Introduction and Antiderivatives	draft	Matthew Mears	27/03/2020 21:13
Johnny's copy of Antiderivatives	draft	Johnny Yi	01/11/2018 01:25
Basic Anti-Derivatives	draft	Johnny Yi	04/12/2018 05:25
Simon's copy of Antiderivatives	draft	Simon Thomas	20/03/2019 14:07
Antiderivatives	draft	Xiaodan Leng	10/07/2019 22:18
INT1	draft	Gagan Aggarwal	03/05/2021 23:16

There are 34 other versions that do you not have access to.

Question statement

Note that

$\begin{eqnarray*}&\int& \;a\;dx&=&ax+C \text{ (for any constant } a)\\ &\int& \;x^n\;dx&=&\frac{x^{n+1}}{n+1}+C,\;\;n \neq -1\\ &\int& \;\sin(ax)\;dx &=& -\frac{1}{a}\cos(ax)+C\\ &\int& \;e^{ax}\;dx &=& \frac{1}{a}e^{ax}+C\\ \end{eqnarray*}$

For each of the following functions $f(x)$ , find $\int f(x) dx$

Use the letter C to represent an unknown constant.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	8
c	integer	4
b	integer	2
d	integer	7
g	integer	2
f	integer	5

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xxxxxxxxxx
 
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Generated value: `integer`

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"Part a)" - prompt
"Part a)" - Correct answer

Advice

Note that

Don't forget to include the unknown constant C.

(a)

$f(x) = \var{a}x - \var{b}$

Noting that $\var{a}x = \var{a}x^1$ and using $\int \;x^n\;dx=\frac{x^{n+1}}{n+1}+C$ with $n=1$ we obtain

$\int f(x)dx = \frac{\var{a}}{2}x^2 - \var{b}x+C= \simplify{{a/2}x^2 - {b}x+C}$

(b)

$f(x) = \frac{1}{\var{c}}+ \frac{2}{\var{d}}x^2 - \frac{3}{\var{f}}x^3$

$\int f(x) dx =\frac{1}{\var{c}}x+ \frac{2}{\var{d}}\frac{x^3}{3} - \frac{3}{\var{f}}\frac{x^4}{4}+C=\simplify{x/{c}+2x^3/(3{d})-3x^4/(4{f})+C}$

(c)

Since $e^\var{g}$ is simply a constant number, we obtain $\int e^\var{g}dx = e^\var{g}x+C$

(d)

$f(x) = e^{\var{g}x}$

Using our rule $\int e^{ax}\;dx = \frac{1}{a}e^{ax}+C$ we immediately obtain

$\int f(x) dx =\frac{1}{\var{g}}e^{\var{g}x}+C$

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This question is used in the following exam:

Integration introduction by Simon Thomas in Calculus.

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Simon Thomas 5 years, 11 months ago

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Simon's copy of Antiderivatives

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 5 years, 11 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

Pattern restriction

Variables

String restrictions

Generated value: `integer`