Shivram's copy of Exam14 - Definite integration

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Shivram Venkat 5 years, 9 months ago

Created this as a copy of Exam14 - Definite integration.

Name	Status	Author	Last Modified
Definite integration using standard integrals	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Exam14 - Definite integration	draft	Michael Proudman	07/12/2021 09:54
Shivram's copy of Exam14 - Definite integration	draft	Shivram Venkat	09/06/2019 18:49
nuExam14 - Definite integration	draft	Michael Proudman	07/12/2021 09:54

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Question statement

Evaluate the following definite integrals.

Give any introductory information the student needs.

			Name	Type	Generated Value

ans1	number	30080.00
ans2	decimal	dec("0.89")
ans3	number	0.00
ans4	number	0.6466
b4	integer	2
b1	integer	6
b2	integer	13
d1	integer	5
s2	integer	-1
s7	integer	1
s6	integer	-1
m4	integer	2
m3	integer	8
m2	integer	9
tol	number	0.01
a1	integer	5
tans4	number	0.6466471676
c1	integer	-4
tans1	number	30080
tans3	number	0
tol1	number	0.0001
p	integer	-2
t	integer	-1
w	integer	0
n4	integer	-1

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xxxxxxxxxx
 
precround(tans1,2)

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Generated value: `number`

30080.00

← Depends on:

tans1

→ Used by:

Advice
"Part a)" → "Unnamed gap" - Minimum accepted value
"Part a)" → "Unnamed gap" - Maximum accepted value

Parts

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1. Gap Number entry
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Part b) Gap-fill
1. Gap Number entry
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Part c) Gap-fill
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Part d) Gap-fill
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$I=\int_1^{\var{b1}}\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\;dx$

$I=\;\;$ Unnamed gap

Input your answer to 2 decimal places.

Advice

$I=\int_1^\var{b1}\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\;dx$
Expand the parentheses to obtain:

$\begin{eqnarray*}I &=& \int_1^\var{b1} \simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\;dx\\ &=&\left[\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\right]_1^\var{b1}\\ &=&\var{tans1}\\ \\&=&\var{ans1}\mbox{ to 2 decimal places} \end{eqnarray*}$

$\begin{eqnarray*}I&=&\int_0^{\var{b2}}\simplify[std]{1/(x+{m2})}\;dx\\ &=&\left[\ln(x+\var{m2})\right]_0^{\var{b2}}\\ &=& \ln(\var{b2+m2})-\ln(\var{m2})\\ &=&\ln\left(\frac{\var{b2+m2}}{\var{m2}}\right)\\ &=&\var{ans2}\mbox{ to 2 decimal places} \end{eqnarray*}$

$I=\int_0^\pi\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\;dx$
We use integration by parts.

Recall that:

$\int u\frac{dv}{dx}\;dx=uv-\int \frac{du}{dx}\;v\;dx$
Here we set

$u=x$ and

$\displaystyle \frac{dv}{dx}=\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$

Hence

$v=\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}$

$\begin{eqnarray*} I&=&\left[\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\right]_0^\pi -\int_0^\pi\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\;dx\\ &=&\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\left[\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\right]_0^\pi\\ &=& \var{ans3}\mbox{ to 2 decimal places} \end{eqnarray*}$
d)

$I=\int_0^{\var{b4}}\simplify[std]{x ^ {m4} * Exp({n4} * x)}\;dx$

Use integration by parts twice with $u=x^2$ and $\displaystyle \frac{dv}{dx}=\simplify[std]{e^({n4}x)}\Rightarrow v = \simplify[std]{1/{n4}e^({n4}x)}$

$\begin{eqnarray*} I&=&\left[\simplify[std]{x^2/{n4}Exp({n4} * x)}\right]_0^{\var{b4}}+\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\ &=&\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\left(\left[\simplify[std]{x/{n4}*e^({n4}*x)}\right]_0^{\var{b4}}+\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\right)\\ &=&\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\left(\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\left[\simplify[std]{1/{n4}*e^({n4}*x)}\right]_0^{\var{b4}}\right)\\ &=&\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\ &=&\var{ans4}\mbox{ to 4 decimal places} \end{eqnarray*}$

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Shivram's copy of Exam14 - Definite integration

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Shivram Venkat 5 years, 9 months ago

Error in variable testing condition

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Shivram's copy of Exam14 - Definite integration

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Shivram Venkat 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`