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2015-06-05T05:01:39Z

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<span class="exam"> Find the antiderivative: <math>\int x^2e^{3x^3}dx.</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
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|This problem requires an advanced rule of integration, namely
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|'''Integration by substitution (''u'' - sub):''' If <math style="vertical-align: -5px">u = g(x)</math>  is a differentiable functions whose range is in the domain of <math style="vertical-align: -5px">f</math>, then
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::<math>\int g'(x)f(g(x)) dx \,=\, \int f(u) du.</math>
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 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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|Use a <math style="vertical-align: 0px">u</math>-substitution with <math style="vertical-align: 0px">u = 3x^3.</math> This means <math style="vertical-align: 0px">du = 9x^2\,dx</math>, or  <math style="vertical-align: -13px">\frac{du}{9x^2}\,=\,dx</math>. After substitution, we have
::<math>\int x^2e^{3x^3}dx\,=\, \int x^2e^u\cdot\frac{du}{9x^2}\,=\,\frac{1}{9}\int e^u\,du.</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
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|From what should be well-known property,
::<math>\frac{1}{9}\int e^u\,du\,=\,\frac{1}{9}\,e^u.</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
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| Now we need to substitute back into our original variables using our original substitution <math style="vertical-align: 0px">u = 3x^3</math> to find  <math style="vertical-align: 0px">e^u\,=\,e^{3x^3}</math>.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 4:  
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|Since this integral is an indefinite integral, we have to remember to add a constant  <math style="vertical-align: 1%">C</math> at the end.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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::<math>\int x^2e^{3x^3}dx\,=\,\frac{1}{9}\,e^{3x^3} + C.</math>
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[[022_Sample_Final_A|'''<u>Return to Sample Exam</u>''']]

022 Sample Final A, Problem 12 - Revision history

MathAdmin: Created page with " Find the antiderivative: \int x^2e^{3x^3}dx. {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations: |- |This..."

MathAdmin: Created page with " Find the antiderivative: $\int x^2e^{3x^3}dx.$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations: |- |This..."