Integration By Parts

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

Find the following indefinite integral.

Input all numbers as fractions or integers and not decimals.

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

Note that if you are entering an expression such as $x*\cos(p)$ for $p$ some expression then you must enter it as x*cos(p).

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
c	integer	-2
b	integer	4
s3	integer	-1
s2	integer	-1
s1	integer	1
a1	integer	2
a2	integer	-9
t	integer	0

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$\displaystyle \int \simplify[std]{{a}x*({t}sin({b}x+{c})+{1-t}*exp({b}x+{c}))} dx = \phantom{{}}$ Unnamed gap

Advice

The formula for integrating by parts is

$\int u\frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx.$

We choose $u = \simplify[std]{{a}x}$ and $\displaystyle \frac{dv}{dx} = \simplify[std]{{t}*sin({b}x+{c})+{1-t}*exp({b}x+{c})}$ .

So $\displaystyle \frac{du}{dx} = \var{a}$ and $v = \simplify[std]{({-t}/{b})*cos({b}x+{c})+{1-t}/{b}*exp({b}x+{c})}$ .

Hence,

$\begin{eqnarray} \int \simplify[std]{{a}x*({t}sin({b}x+{c})+{1-t}*exp({b}x+{c})) } dx &=& uv - \int v \frac{du}{dx} dx \\ &=& \simplify[std]{({a}/{b})*x*(({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}))-{a}/{b}int (({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}),x)} \\ &=& \simplify[std]{({a}/{b})*x*(({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c})) + ({t*a}/{b^2})*sin({b}x+{c}) -{(1-t)*a}/{b^2}*exp({b}x+{c})+ C} \end{eqnarray}$

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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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Integration By Parts

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`