Difference between revisions of "009B Sample Final 3, Problem 5"

From Grad Wiki
Jump to navigation Jump to search
Line 35: Line 35:
 
|Therefore, we have
 
|Therefore, we have
 
|-
 
|-
−
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\int.</math>
+
|&nbsp; &nbsp; &nbsp; &nbsp;<math>\int x\cos(x)~dx=x\sin x -\int \sin x~dx.</math>
 
|}
 
|}
  
Line 41: Line 41:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
+
|Then, we integrate to get
 
|-
 
|-
−
|  
+
|&nbsp; &nbsp; &nbsp; &nbsp; <math> \int x\cos(x)~dx=x\sin x +\cos x+C.</math>
 
|-
 
|-
 
|
 
|
Line 90: Line 90:
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)'''  
+
|&nbsp; &nbsp;'''(a)''' &nbsp; &nbsp;<math>x\sin x +\cos x+C</math>
 
|-
 
|-
 
|'''(b)'''  
 
|'''(b)'''  

Revision as of 15:39, 28 February 2017

Find 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 x\cos(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 \sin^3(x)\cos^2(x)~dx}

Foundations:  
1. Integration by parts tells us that
        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 u~dv=uv-\int v~du.}
2. Since  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 \sin^2x+\cos^2x=1,}   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 \sin^2x=1-\cos^2x.}


Solution:

(a)

Step 1:  
To calculate this integral, 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  
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=\sin x.}
Therefore, 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 x\cos(x)~dx=x\sin x -\int \sin x~dx.}
Step 2:  
Then, we integrate 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 x\cos(x)~dx=x\sin x +\cos x+C.}

(b)

Step 1:  
Step 2:  

(c)

Step 1:  
Step 2:  


Final Answer:  
   (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 x\sin x +\cos x+C}
(b)
(c)

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009B_Sample_Final_3,_Problem_5&oldid=2902"