Quiz:Integration by parts: Difference between revisions
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- <math>x^2f(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other. | - <math>x^2f(x^2)</math> has an antiderivative expressible in terms of <math>F</math>. The integration problems for the other two functions are equivalent to each other. | ||
- All the integration problems are equivalent to each other, but none has a guaranteed expression in terms of <math>f</math> and <math>F</math>. | - All the integration problems are equivalent to each other, but none has a guaranteed expression in terms of <math>f</math> and <math>F</math>. | ||
</quiz> | </quiz> | ||
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{Suppose <math>p</math> is a polynomial function. In order to find the indefinite integral for a function of the form <math>x \mapsto p(x)\exp(x)</math>, the general strategy, which always works, is to take <math>p(x)</math> as the part to differentiate and <math>\exp(c)</math> as the part to integrate, and keep repeating the process. Which of the following is the best explanation for why this strategy works? | {Suppose <math>p</math> is a polynomial function. In order to find the indefinite integral for a function of the form <math>x \mapsto p(x)\exp(x)</math>, the general strategy, which always works, is to take <math>p(x)</math> as the part to differentiate and <math>\exp(c)</math> as the part to integrate, and keep repeating the process. Which of the following is the best explanation for why this strategy works? | ||
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- <math>\exp</math> can be repeatedly differentiated and polynomials can be repeatedly integrated, giving polynomials all the way. | - <math>\exp</math> can be repeatedly differentiated and polynomials can be repeatedly integrated, giving polynomials all the way. | ||
+ <math>\exp</math> can be repeatedly integrated and polynomials can be repeatedly differentiated, eventually becoming zero. | + <math>\exp</math> can be repeatedly integrated and polynomials can be repeatedly differentiated, eventually becoming zero. | ||
- <math>\exp</math> and polynomials can both be repeatedly differentiated. | - <math>\exp</math> and polynomials can both be repeatedly differentiated. | ||
- <math>\exp</math> and polynomials can both be repeatedly integrated. | - <math>\exp</math> and polynomials can both be repeatedly integrated. | ||
{Suppose <math>a</math> and <math>b</math> are real numbers that are not positive integers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int x^ae^x \, dx</math> and <math>\int x^be^x \, dx</math> to be equivalent? | |||
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- <math>a + b</math> is an integer. | |||
+ <math>a - b</math> is an integer. | |||
- <math>ab</math> is an integer. | |||
- <math>a/b</math> is an integer. | |||
{Suppose <math>a</math> and <math>b</math> are real numbers that are not positive integers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int x^ae^x \, dx</math> and <math>\int e^{x^b} \, dx</math> to be equivalent? | |||
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- <math>a + b = 1</math> | |||
- <math>a - b = 1</math> | |||
+ <math>ab = 1</math> | |||
- <math>a/b = 1</math> | |||
</quiz> | </quiz> |
Revision as of 03:16, 20 February 2012
For background, see integration by parts.
Key observations
Equivalence of integration problems
Specific integration types