022 Exam 2 Sample A, Problem 4 - Revision history

MathAdmin at 23:22, 15 May 2015

2015-05-15T23:22:10Z

MathAdmin at 23:21, 15 May 2015

2015-05-15T23:21:52Z

MathAdmin at 23:21, 15 May 2015

2015-05-15T23:21:37Z

MathAdmin at 23:12, 15 May 2015

2015-05-15T23:12:38Z

MathAdmin at 22:47, 15 May 2015

2015-05-15T22:47:34Z

MathAdmin at 18:03, 15 May 2015

2015-05-15T18:03:16Z

MathAdmin: Created page with " Find the antiderivative of $\int (3x+2)^4\,dx.$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;"..."

2015-05-15T16:57:23Z

Created page with "<span class="exam"> Find the antiderivative of <math style="vertical-align: -45%">\int (3x+2)^4\,dx.</math> {| class="mw-collapsible mw-collapsed" style = "text-align:left;"..."

New page

<span class="exam"> Find the antiderivative of <math style="vertical-align: -45%">\int (3x+2)^4\,dx.</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|This problem requires two rules of integration, integration by substitution '''(U - sub)''' and the power rule.
|-
|
|-
||Additionally, we will need our power rule for integration:
|-
|
::<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>,
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
||Use a U-substitution with <math>u = 3x + 2.</math> This means <math>du = 3 dx</math>, and after substitution we have
::<math>\int \left(3x + 2\right)^4 dx = \int u^4 du</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|We can no apply the power rule for integration:
::<math>\int u^4 du = \frac{u^5}{5}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  
|-
| Since our original function is a function of x, we must substitute x back into the result from problem 2:
|-
|
::<math>\frac{u^5}{5} = \frac{(3x + 2)^5}{5}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 4:  
|-
| As will all indefinite integrals, don't forget the ''' "+C" ''' at the end.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|<math>\int \left(3x + 2\right)^5 dx\,=\, \frac{(3x + 2)^5}{5} + C</math>
|}

[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

@@ Line 22: / Line 22: @@
 !Step 1: &nbsp;
 |-
-|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
+|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: -21%">dx=du/3</math>. After substitution we have
 ::<math>\int \left(3x + 2\right)^4 \,dx  \,=\, \int u^4 \,\frac{du}{3}\,=\,\frac{1}{3}\int u^4\,du.</math>
 |}

@@ Line 14: / Line 14: @@
 |-
+|
-::<math style="vertical-align: -70%;">\int x^n dx \,=\, \frac{x^{n + 1}}{n + 1}+C,</math>&thinsp; for <math style="vertical-align: -20%;">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: -23%;">n\neq 0</math>.
 |}

@@ Line 14: / Line 14: @@
 |-
+|
-::<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>.
+::<math style="vertical-align: -70%;">\int x^n dx \,=\, \frac{x^{n + 1}}{n + 1}+C,</math>&thinsp; for <math style="vertical-align: -20%;">n\neq 0</math>.
 |}

@@ Line 6: / Line 6: @@
 |This problem requires three rules of integration.  In particular, you need
 |-
-|'''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>u = g(x)</math> is a differentiable functions whose range is in the domain of <math>f</math>, then
 |-
 |<math>\int g'(x)f(g(x)) dx = \int f(u) du.</math>
 |-
-|We also need the derivative of the natural log since we will recover natural log from integration:
+|We also need our power rule for integration:
 |-
+|

@@ Line 4: / Line 4: @@
 !Foundations: &nbsp;
 |-
-|This problem requires two rules of integration, integration by substitution '''(U - sub)''' and the power rule.
+|This problem requires three rules of integration.  In particular, you need
 |-
+|'''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
 |-
-||Additionally, we will need our power rule for integration:
+||Finally, we will need our power rule for integration:
 |-
+|

@@ Line 6: / Line 6: @@
 |This problem requires three rules of integration.  In particular, you need
 |-
-|'''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
 |-
 |<math>\int g'(x)f(g(x)) dx = \int f(u) du.</math>
 |-
 |We also need our power rule for integration:
 |-
+|
-::<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>.
 |}
@@ Line 21: / Line 22: @@
 !Step 1: &nbsp;
 |-
-||Use a U-substitution with <math>u = 3x + 2.</math> This means <math>du = 3 dx</math>, and after substitution we have
+|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>
 |}
@@ Line 29: / Line 30: @@
 |-
 |We can no apply the power rule for integration:
-::<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>
 |}
@@ Line 35: / Line 36: @@
 !Step 3: &nbsp;
 |-
-| Since our original function is a function of x, we must substitute x back into the result from problem 2:
+|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:
 |-
+|
-::<math>\frac{u^5}{5} = \frac{(3x + 2)^5}{5}</math>
+::<math>\frac{u^5}{5} \,=\, \frac{(3x + 2)^5}{5}.</math>
 |}
@@ Line 44: / Line 45: @@
 !Step 4: &nbsp;
 |-
-| As will all indefinite integrals, don't forget the  ''' "+C" ''' at the end.
+| As will all indefinite integrals, don't forget the constant&thinsp; <math style="vertical-align: 0%">C</math> at the end.
 |}
@@ Line 50: / Line 51: @@
 !Final Answer: &nbsp;
 |-
-|<math>\int \left(3x + 2\right)^5 dx\,=\, \frac{(3x + 2)^5}{5} + C</math>
 |}
 [[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

022 Exam 2 Sample A, Problem 4 - Revision history

MathAdmin at 23:22, 15 May 2015

MathAdmin at 23:21, 15 May 2015

MathAdmin at 23:21, 15 May 2015

MathAdmin at 23:12, 15 May 2015

MathAdmin at 22:47, 15 May 2015

MathAdmin at 18:03, 15 May 2015

MathAdmin: Created page with " Find the antiderivative of \int (3x+2)^4\,dx. {| class="mw-collapsible mw-collapsed" style = "text-align:left;"..."

MathAdmin: Created page with " Find the antiderivative of $\int (3x+2)^4\,dx.$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;"..."