Integration: Indefinite integral with 1/ (ax+b)^n

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Johnny Yi 6 years, 3 months ago

Created this as a copy of Luis's copy of Integration: Indefinite integral.

Name	Status	Author	Last Modified
Integration: Indefinite integral	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Luis's copy of Integration: Indefinite integral	draft	Luis Hernandez	30/11/2018 18:29
Integration: Indefinite integral with 1/ (ax+b)^n	draft	Johnny Yi	19/12/2018 01:24

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Question statement

Find the following indefinite integral.

Input the constant of integration as $C$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	5
b	integer	5
d	integer	5
n	integer	4

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Generated value: `integer`

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

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$ .

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

$\simplify[std]{{b}/(({a}*x+{d})^{n})} = \simplify[std]{{b}/(y^{n})}.$

Now,

$\int \simplify[std]{{b}/({a}*x+{d})^{n}} dx = \int \simplify[std]{{b}/(y^{n})} \frac{dx}{dy} dy.$

Rearrange $y = \simplify[std]{{a}x+{d}}$ to get $\displaystyle x = \simplify[std]{(y-{b})/{a}}$ , and hence $\displaystyle\frac{dx}{dy} = \frac{1}{\var{a}}$ .

$\displaystyle \int \frac{1}{y^n} dx = -\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:

$\int \simplify[std]{{b}/(y^{n})} \frac{dx}{dy} dy = \simplify[std]{{b}/({n-1}*y^{n-1})} \cdot \frac{1}{\var{a}} + C = \simplify[std]{(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}.$

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Integration: Indefinite integral with 1/ (ax+b)^n

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Johnny Yi 6 years, 3 months ago

Error in variable testing condition

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Generated value: `integer`

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Integration: Indefinite integral with 1/ (ax+b)^n

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Johnny Yi 6 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`