Difference between revisions of "022 Exam 2 Sample B, Problem 7"
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|(a) Use a ''u''-substitution with <math style="vertical-align: -8%">u = 3x^2 + 1.</math> This means <math style="vertical-align: 0%">du = 6x\,dx</math>, or <math style="vertical-align: -20%">dx=du/6</math>. After substitution we have | |(a) Use a ''u''-substitution with <math style="vertical-align: -8%">u = 3x^2 + 1.</math> This means <math style="vertical-align: 0%">du = 6x\,dx</math>, or <math style="vertical-align: -20%">dx=du/6</math>. After substitution we have | ||
− | ::<math>\int x e^{3x^2+1} | + | ::<math>\int x e^{3x^2+1} dx = \frac{1}{6} \int e^{u} du. </math> |
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|(b) We need to use the power rule to find that <math>\int_2^5 4x - 5 \, dx = 2x^2 - 5x \Bigr|_2^5</math> | |(b) We need to use the power rule to find that <math>\int_2^5 4x - 5 \, dx = 2x^2 - 5x \Bigr|_2^5</math> |
Revision as of 17:32, 15 May 2015
Find the antiderivatives:
- (a)
- (b)
Foundations: |
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This problem requires Integration by substitution (u - sub): If is a differentiable functions whose range is in the domain of , then |
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We also need our power rule for integration: |
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Solution:
Step 1: |
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(a) Use a u-substitution with This means , or . After substitution we have
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(b) We need to use the power rule to find that |
Step 2: |
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(a)
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(b) We just need to evaluate at the endpoints to finish the problem:
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Step 3: |
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(a) Now we need to substitute back into our original variables using our original substitution |
to find |
Step 4: |
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Since this integral is an indefinite integral we have to remember to add a constant at the end. |
Final Answer: |
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(a)
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(b) 27 |