Indefinite integral

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Bill Foster

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Bill Foster 55 years, 3 months ago

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Question statement

Find the following indefinite integral.

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

Input the constant of integration as $C$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	4
c	integer	-4
b	integer	-4
d	integer	1
s2	integer	-1
s1	integer	-1
m	integer	-1
n	integer	5
r	integer	-3

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$I=\displaystyle \int \simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$

You are given that

$I=\simplify[std]{g(x)*({a}x+{d})^{1-n}}+C$ for a polynomial

$g(x)$ . You have to find

$g(x)$ .

$g(x)=\;$ Unnamed gap

Remember to input all numbers as integers or fractions.

Click on Show steps to get help if you need it. You will lose 1 mark by doing so.

Advice

Let $y = \simplify[std]{{a}*x+{d}}$ .

Then $\displaystyle x=\frac{1}{\var{a}}\simplify[std]{(y-{d})}$ and so we have the numerator $\simplify[std]{{b}*x+{c}}$ becomes in terms of $y$ :

$\displaystyle \simplify[std]{{b}*x+{c} = {b}*1/{a}*(y-{d})+{c}= {m}y+{r}}$ and so

$\displaystyle \simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} = \simplify[std]{({m}*y+{r})/(y^{n})={m}/y^{n-1}+{r}/y^{n}}$

Now,

$\int \simplify[std]{({b}x+{c})/({a}*x+{d})^{n}} dx = \int \left(\simplify[std]{{m}/y^{n-1}+{r}/y^{n}} \right)\frac{dx}{dy} dy$

Since $\displaystyle x = \simplify[std]{(y-{d})/{a}}$ then $\displaystyle \frac{dx}{dy} = \frac{1}{\var{a}}$ .

We can now calculate the desired integral:

$\begin{eqnarray*} \int \left(\simplify[std]{{m}/y^{n-1}+{r}/y^{n}}\right) \frac{dx}{dy} dy &=&\frac{1}{\var{a}}\left(\int \simplify[std]{{m}/y^{n-1}}\;dy+\int \simplify[std]{{r}/y^{n}}\;dy \right)\\ &=&\frac{1}{\var{a}}\left(\simplify[std]{{-m}/({n-2}*y^{n-2})+ {-r}/({n-1}*y^{n-1})}\right) + C \\ &=& \simplify[std]{(-{m})/({a*(n-2)}*({a}*x+{d})^{n-2})+(-{r})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}\\ &=&\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{a*(n-2)})*({a}x+{d})-{r}/({a*(n-1)}))}\\ &=&\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a})} \end{eqnarray*}$
Hence

$g(x)=\simplify[std]{({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a}}$

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Bill Foster 55 years, 3 months ago

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Generated value: `integer`

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Indefinite integral

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 3 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`