Difference between revisions of "009B Sample Midterm 3, Problem 3"
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!Step 1: | !Step 1: | ||
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| − | | | + | |Again, we proceed using u substitution. Let <math>u=\cos(x)</math>. Then, <math>du=-\sin(x)dx</math>. |
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| − | | | + | |Since this is a definite integral, we need to change the bounds of integration. |
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| − | | | + | |We have <math>u_1=\cos\bigg(-\frac{\pi}{4}\bigg)=\frac{\sqrt{2}}{2}</math> and <math>u_2=\cos\bigg(\frac{\pi}{4}\bigg)=\frac{\sqrt{2}}{2}</math>. |
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!Step 2: | !Step 2: | ||
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| − | | | + | |So, we get |
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| − | | | + | |<math>\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2(x)\sin (x)dx=\int_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{2}}{2}} -u^2=\left.\frac{-u^3}{3}\right|_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{2}}{2}}=0</math> |
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|'''(a)''' <math>\frac{-1}{3}\cos(x^3)+C</math> | |'''(a)''' <math>\frac{-1}{3}\cos(x^3)+C</math> | ||
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| − | |'''(b)''' | + | |'''(b)''' <math>0</math> |
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[[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 12:46, 31 January 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 x^2\sin (x^3) 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{\pi}{4}}^{\frac{\pi}{4}} \cos^2(x)\sin (x)dx}
| Foundations: |
|---|
| u substitution |
Solution:
(a)
| Step 1: |
|---|
| We proceed using 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=x^3} . 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=3x^2dx} . |
| 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^2\sin (x^3) dx=\int \frac{\sin(u)}{3}du} |
| Step 2: |
|---|
| 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^2\sin (x^3) dx=\frac{-1}{3}\cos(u)+C=\frac{-1}{3}\cos(x^3)+C} |
(b)
| Step 1: |
|---|
| Again, we proceed using 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} . |
| Since this is a definite integral, we need to change the bounds of integration. |
| 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 u_1=\cos\bigg(-\frac{\pi}{4}\bigg)=\frac{\sqrt{2}}{2}} and . |
| Step 2: |
|---|
| So, we 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_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2(x)\sin (x)dx=\int_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{2}}{2}} -u^2=\left.\frac{-u^3}{3}\right|_{\frac{\sqrt{2}}{2}}^{\frac{\sqrt{2}}{2}}=0} |
| 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 \frac{-1}{3}\cos(x^3)+C} |
| (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 0} |