Difference between revisions of "009B Sample Final 1, Problem 4"

From Grad Wiki
Jump to navigation Jump to search
Line 11: Line 11:
 
!Foundations:    
 
!Foundations:    
 
|-
 
|-
−
|
+
|Review <math>u</math>-substitution and
 +
|-
 +
|Integration by parts
 
|}
 
|}
  
Line 21: Line 23:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|
+
|We first distribute to get <math>\int e^x(x+\sin(e^x))~dx=\int e^xx~dx+\int e^x\sin(e^x)~dx</math>.
 
|-
 
|-
−
|
+
|Now, for the first integral on the right hand side of the last equation, we use integration by parts.
 
|-
 
|-
−
|
+
|Let <math>u=x</math> and <math>dv=e^xdx</math>. Then, <math>du=dx</math> and <math>v=e^x</math>. So, we have
 
|-
 
|-
−
|
+
|<math>\int e^x(x+\sin(e^x))~dx=\bigg(xe^x-\int e^x~dx \bigg)+\int e^x\sin(e^x)~dx=xe^x-e^x+\int e^x\sin(e^x)~dx</math>
 
|}
 
|}
  

Revision as of 07:33, 2 February 2016

Compute the following integrals.

a) 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 e^x(x+\sin(e^x))~dx}

b) 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 \frac{2x^2+1}{2x^2+x}~dx}

c) 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 \sin^3x~dx}


Foundations:  
Review 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} -substitution and
Integration by parts

Solution:

(a)

Step 1:  
We first distribute to get 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 e^x(x+\sin(e^x))~dx=\int e^xx~dx+\int e^x\sin(e^x)~dx} .
Now, for the first integral on the right hand side of the last equation, we use integration by parts.
Let 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=x} and 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 dv=e^xdx} . 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 du=dx} and 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 v=e^x} . So, 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 e^x(x+\sin(e^x))~dx=\bigg(xe^x-\int e^x~dx \bigg)+\int e^x\sin(e^x)~dx=xe^x-e^x+\int e^x\sin(e^x)~dx}
Step 2:  

(b)

Step 1:  
Step 2:  
Step 3:  

(c)

Step 1:  
Step 2:  
Final Answer:  
(a)
(b)
(c)

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009B_Sample_Final_1,_Problem_4&oldid=1301"