Quiz:Integration by parts: Difference between revisions
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== | ==Equivalence of integration problems== | ||
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- Knowledge of an antiderivative for <math>x \mapsto x^2f(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>. | - Knowledge of an antiderivative for <math>x \mapsto x^2f(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>. | ||
+ Knowledge of an antiderivative for <math>x \mapsto xf(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>. | + Knowledge of an antiderivative for <math>x \mapsto xf(x)</math> is equivalent to knowledge of an antiderivative for <math>F</math>. | ||
{Suppose <math>f</math> is a function with a known antiderivative <math>F</math>. Which of the following integration problems is ''not'' equivalent to the others? | |||
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- <math>\int f(\sqrt{x}) \, dx</math> | |||
- <math>\int xf(x) \, dx</math> | |||
+ <math>\int f(x^2) \, dx</math> | |||
- <math>\int F(x) \, dx</math> | |||
{In order to find the indefinite integral for a function of the form <math>x \mapsto p(x)\sin x</math>, the general strategy, which always works, is to take <math>p(x)</math> as the part to differentiate and <math>\sin x</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>\sin</math> can be repeatedly differentiated and polynomials can be repeatedly integrated, giving polynomials all the way. | |||
+ <math>\sin</math> can be repeatedly integrated and polynomials can be repeatedly differentiated, eventually becoming zero. | |||
- <math>\sin</math> and polynomials can both be repeatedly differentiated. | |||
- <math>\sin</math> and polynomials can both be repeatedly integrated. | |||
{Suppose we know the first three antiderivatives for <math>f</math>, i.e., we have explicit expressions for an antiderivative of <math>f</math>, an antiderivative of that antiderivative, and an antiderivative of the antiderivative of the antiderivative. What is the largest nonnegative integer <math>k</math> for which this guarantees us an expression for an antiderivative of <math>x \mapsto x^kf(x)</math>? | |||
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{Suppose we know the first three antiderivatives for <math>f</math>, i.e., we have explicit expressions for an antiderivative of <math>f</math>, an antiderivative of that antiderivative, and an antiderivative of the antiderivative of the antiderivative. What is the largest nonnegative integer <math>k</math> for which this guarantees us an expression for an antiderivative of <math>x \mapsto f(x^{1/k})</math>? For simplicity, assume that we are only considering <math>x > 0</math>. | |||
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{Suppose <math>f</math> has a known antiderivative <math>F</math>. Consider the problems of integrating <math>f(x^2), xf(x^2), x^2f(x^2)</math>. What can we say about the relation between these problems? | |||
- All of these have antiderivatives expressible in terms of <math>f</math> and <math>F</math>. | |||
- <math>f(x^2)</math> has an antiderivative expressible in terms of <math>f</math> and <math>F</math>. The integration problems for the other two functions are equivalent to each other. | |||
+ <math>xf(x^2)</math> has an antiderivative expressible in terms of <math>f</math> and <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> and <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>. | |||
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Revision as of 00:12, 29 December 2011
Key observations
Equivalence of integration problems