Difference between revisions of "009B Sample Final 3, Problem 5"
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|To calculate this integral, we use integration by parts. | |To calculate this integral, we use integration by parts. | ||
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| − | |Let <math>u=x</math> and <math>dv=\cos xdx.</math> | + | |Let <math style="vertical-align: 0px">u=x</math> and <math style="vertical-align: 0px">dv=\cos xdx.</math> |
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| − | |Then, <math>du=dx</math> and <math>v=\sin x.</math> | + | |Then, <math style="vertical-align: 0px">du=dx</math> and <math style="vertical-align: -1px">v=\sin x.</math> |
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|Therefore, we have | |Therefore, we have | ||
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'''(b)''' | '''(b)''' | ||
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
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| '''(a)''' <math>x\sin x +\cos x+C</math> | | '''(a)''' <math>x\sin x +\cos x+C</math> | ||
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| − | |'''(b)''' | + | | '''(b)''' |
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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 15:51, 28 February 2017
Find the following integrals.
(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 \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: |
|---|
| 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) |