Difference between revisions of "009B Sample Midterm 1, Problem 3"
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!Step 1: | !Step 1: | ||
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| − | | | + | |We proceed using integration by parts. Let <math>u=x^2</math> and <math>dv=e^xdx</math>. Then, <math>du=2xdx</math> and <math>v=e^x</math>. |
| + | |- | ||
| + | |Therefore, we have | ||
|- | |- | ||
| − | | | + | |<math>\int x^2 e^xdx=x^2e^x-\int 2xdx</math> |
|} | |} | ||
| Line 25: | Line 27: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |Now, we need to use integration by parts again. Let <math>u=2x</math> and <math>dv=e^xdx</math>. Then, <math>du=2dx</math> and <math>v=e^x</math>. |
| + | |- | ||
| + | |Therefore, we have | ||
|- | |- | ||
| − | | | + | |<math>\int x^2 e^xdx=x^2e^x-\bigg(2xe^x-\int 2e^xdx\bigg)=x^2e^x-2xe^x+2e^x+C</math> |
|} | |} | ||
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!Final Answer: | !Final Answer: | ||
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| − | |'''(a)''' | + | |'''(a)''' <math>x^2e^x-2xe^x+2e^x+C</math> |
|- | |- | ||
|'''(b)''' | |'''(b)''' | ||
|} | |} | ||
[[009B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] | [[009B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 11:08, 31 January 2016
Evaluate the indefinite and definite 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 e^xdx}
- 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_{1}^{e} x^3\ln x~dx}
| Foundations: |
|---|
| Review integration by parts |
Solution:
(a)
| Step 1: |
|---|
| We proceed using 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^2} 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=2xdx} 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} . |
| 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 e^xdx=x^2e^x-\int 2xdx} |
| Step 2: |
|---|
| Now, we need to use integration by parts again. 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=2x} 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=2dx} 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} . |
| 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 e^xdx=x^2e^x-\bigg(2xe^x-\int 2e^xdx\bigg)=x^2e^x-2xe^x+2e^x+C} |
(b)
| Step 1: |
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| Step 2: |
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| Final Answer: |
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| (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^2e^x-2xe^x+2e^x+C} |
| (b) |