Difference between revisions of "022 Exam 2 Sample A, Problem 4"

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|This problem requires three rules of integration.  In particular, you need
 
|This problem requires three rules of integration.  In particular, you need
 
|-
 
|-
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|'''Integration by substitution (u - sub):''' If <math>u = g(x)</math> is a differentiable functions whose range is in the domain of <math>f</math>, then
+
|'''Integration by substitution (''u'' - sub):''' If <math style="vertical-align: -25%">u = g(x)</math>&thinsp; is a differentiable functions whose range is in the domain of <math style="vertical-align: -20%">f</math>, then
 
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|-
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|<math>\int g'(x)f(g(x)) dx = \int f(u) du.</math>
+
|
 +
::<math>\int g'(x)f(g(x)) dx = \int f(u) du.</math>
 
|-
 
|-
 
|We also need our power rule for integration:
 
|We also need our power rule for integration:
 
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|-
 
|
 
|
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::<math style="vertical-align: -21%;">\int x^n dx \,=\, \frac{x^{n + 1}}{n + 1},</math> for <math style="vertical-align: -25%;">n\neq 0</math>,
+
::<math style="vertical-align: -70%;">\int x^n dx \,=\, \frac{x^{n + 1}}{n + 1}+C,</math>&thinsp; for <math style="vertical-align: -25%;">n\neq 0</math>.
 
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!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
|-
 
|-
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||Use a U-substitution with <math>u = 3x + 2.</math> This means <math>du = 3 dx</math>, and after substitution we have
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|Use a ''u''-substitution with <math style="vertical-align: -8%">u = 3x + 2.</math> This means <math style="vertical-align: 0%">du = 3\,dx</math>, or <math style="vertical-align: -23%">dx=du/3</math>. After substitution we have
−
::<math>\int \left(3x + 2\right)^4 dx  = \int u^4 du</math>
+
::<math>\int \left(3x + 2\right)^4 \,dx  \,=\, \int u^4 \,\frac{du}{3}\,=\,\frac{1}{3}\int u^4\,du.</math>
 
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|-
 
|-
 
|We can no apply the power rule for integration:
 
|We can no apply the power rule for integration:
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::<math>\int u^4 du = \frac{u^5}{5}</math>
+
::<math>\frac{1}{3}\int u^4\,du \,=\, \frac{1}{3}\cdot\frac{u^5}{5}\,=\,\frac{u^5}{15}.</math>
 
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|}
  
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!Step 3: &nbsp;
 
!Step 3: &nbsp;
 
|-
 
|-
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| Since our original function is a function of x, we must substitute x back into the result from problem 2:
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|Since our original function is a function of <math style="vertical-align: 0%">x</math>, we must substitute <math style="vertical-align: 0%">x</math> back into the result from step 2:
 
|-
 
|-
 
|
 
|
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::<math>\frac{u^5}{5} = \frac{(3x + 2)^5}{5}</math>
+
::<math>\frac{u^5}{5} \,=\, \frac{(3x + 2)^5}{5}.</math>
 
|}
 
|}
  
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!Step 4: &nbsp;
 
!Step 4: &nbsp;
 
|-
 
|-
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| As will all indefinite integrals, don't forget the ''' "+C" ''' at the end.
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| As will all indefinite integrals, don't forget the constant&thinsp; <math style="vertical-align: 0%">C</math> at the end.
 
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!Final Answer: &nbsp;
 
!Final Answer: &nbsp;
 
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|<math>\int \left(3x + 2\right)^5 dx\,=\, \frac{(3x + 2)^5}{5} + C</math>
+
|
 +
::<math>\int \left(3x + 2\right)^4\,dx\,=\, \frac{(3x + 2)^5}{15} + C.</math>
 
|}
 
|}
  
 
[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]
 
[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

Revision as of 15:12, 15 May 2015

Find the antiderivative of

Foundations:  
This problem requires three rules of integration. In particular, you need
Integration by substitution (u - sub): If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = g(x)}   is a differentiable functions whose range is in the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , then
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int g'(x)f(g(x)) dx = \int f(u) du.}
We also need our power rule for integration:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^n dx \,=\, \frac{x^{n + 1}}{n + 1}+C,}   for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\neq 0} .

 Solution:

Step 1:  
Use a u-substitution with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = 3x + 2.} This means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle du = 3\,dx} , or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx=du/3} . After substitution we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \left(3x + 2\right)^4 \,dx \,=\, \int u^4 \,\frac{du}{3}\,=\,\frac{1}{3}\int u^4\,du.}
Step 2:  
We can no apply the power rule for integration:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3}\int u^4\,du \,=\, \frac{1}{3}\cdot\frac{u^5}{5}\,=\,\frac{u^5}{15}.}
Step 3:  
Since our original function is a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , we must substitute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} back into the result from step 2:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{u^5}{5} \,=\, \frac{(3x + 2)^5}{5}.}
Step 4:  
As will all indefinite integrals, don't forget the constant  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} at the end.
Final Answer:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \left(3x + 2\right)^4\,dx\,=\, \frac{(3x + 2)^5}{15} + C.}

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