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

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!Step 1:    
 
!Step 1:    
 
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|First, we use the identity <math>\sin^2 x=1-\cos^2 x</math> to get
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\int \sin^3(x)\cos^2(x)~dx} & = & \displaystyle{\int \sin^2 x (\cos^2 x) \sin x~dx}\\
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&&\\
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& = & \displaystyle{\int (1-\cos^2 x)(\cos^2 x) \sin x~dx.}
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\end{array}</math>
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now, we use <math>u</math>-substitution.
 
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|Let <math>u=\cos(x)</math>. Then, <math>du=-\sin(x)dx</math> and <math>-du=\sin(x)dx.</math>
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|Therefore, we have
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
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\displaystyle{\int \sin^3(x)\cos^2(x)~dx} & = & \displaystyle{\int (-1)(1-u^2)u^2~du}\\
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&&\\
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& = & \displaystyle{\int u^4-u^2~du}\\
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&&\\
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& = & \displaystyle{\frac{u^5}{5}-\frac{u^3}{3}+C}\\
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&&\\
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& = & \displaystyle{\frac{\cos^5 x}{5}-\frac{\cos^3 x}{3}+C.}
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\end{array}</math>
 
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|&nbsp; &nbsp;'''(a)''' &nbsp; &nbsp;<math>x\sin x +\cos x+C</math>
 
|&nbsp; &nbsp;'''(a)''' &nbsp; &nbsp;<math>x\sin x +\cos x+C</math>
 
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|&nbsp; &nbsp;'''(b)''' &nbsp; &nbsp;  
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|&nbsp; &nbsp;'''(b)''' &nbsp; &nbsp;<math>\frac{\cos^5 x}{5}-\frac{\cos^3 x}{3}+C</math>
 
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[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 16:07, 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  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=\cos 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=\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:  
First, we use the identity 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^2 x=1-\cos^2 x} 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 \begin{array}{rcl} \displaystyle{\int \sin^3(x)\cos^2(x)~dx} & = & \displaystyle{\int \sin^2 x (\cos^2 x) \sin x~dx}\\ &&\\ & = & \displaystyle{\int (1-\cos^2 x)(\cos^2 x) \sin x~dx.} \end{array}}
Step 2:  
Now, we use 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.
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=\cos(x)} . 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=-\sin(x)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 -du=\sin(x)dx.}
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 \begin{array}{rcl} \displaystyle{\int \sin^3(x)\cos^2(x)~dx} & = & \displaystyle{\int (-1)(1-u^2)u^2~du}\\ &&\\ & = & \displaystyle{\int u^4-u^2~du}\\ &&\\ & = & \displaystyle{\frac{u^5}{5}-\frac{u^3}{3}+C}\\ &&\\ & = & \displaystyle{\frac{\cos^5 x}{5}-\frac{\cos^3 x}{3}+C.} \end{array}}


Final Answer:  
   (a)    
   (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 \frac{\cos^5 x}{5}-\frac{\cos^3 x}{3}+C}

Return to Sample Exam

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