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| || Applying integration by parts twice, we get <math>\int e^x \sin x \,dx = e^x (\sin x - \cos x) - \int e^x\sin x \, dx</math>. Now, rearrange and simplify. | | || Applying integration by parts twice, we get <math>\int e^x \sin x \,dx = e^x (\sin x - \cos x) - \int e^x\sin x \, dx</math>. Now, rearrange and simplify. |
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| {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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| |type="()"}
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| - <math>a + b</math> is an integer.
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| + <math>a - b</math> is an integer.
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| || For simplicity, assume <math>a < b</math> (the process works exactly the same way in reverse if <matH>b < a</math>). Start with the integral <math>\int x^be^x \, dx</math>. Now apply integration by parts taking <math>e^x</math> as the part to integrate and <math>x^b</math> as the part to differentiate. After one application of integration by parts, we need to integrate <math>x^{b-1}e^x</math>. Proceed in the way and we see that we get the integrations of <math>x^be^x, x^{b-1}e^x, x^{b-2}e^x, \dots</math>. If <math>a,b</math> differ by an integer, then after finitely many steps, we will land up with <math>\int x^a e^x\, dx</math>.
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| - <math>ab</math> is an integer.
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| - <math>a/b</math> is an integer.
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| {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? Assume we are working with <math>x > 0</math>, so any real power of <math>x</matH> makes sense.
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| |type="()"}
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| - <math>a + b = 1</math>
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| - <math>a - b = 1</math>
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| + <math>ab = 1</math>
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| || Using integration by parts once, we can convert <math>\int x^a e^x\, dx</math> to <math>\int ax^{a-1} e^x \, dx</math>. Now, put <math>u = x^a</math>. Then <math>x = u^{1/a}</math>, and <math>du = ax^{a-1} \, dx</math>. So, we get that the integral is <math>\int e^{u^{1/a}} \, du</math>. Replace the dummy variable <math>u</math> by the dummy variable <math>x</math>, to obtain <math>\int e^{x^{1/a}} \, dx</math>, which is <math>\int e^{x^b} \, dx</math> by the assumption that <math>b = 1/a</math>.
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| - <math>a/b = 1</math>
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| {Suppose <math>a</math> and <math>b</math> are positive real numbers. Which of the following is a ''sufficient'' condition for the integration problems <math>\int e^{x^a} \, dx</math> and <math>\int e^{x^b} \, dx</math> to be equivalent? Assume we are working with <math>x > 0</math>, so any real power of <math>x</matH> makes sense.
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| |type="()"}
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| - <math>1/a + 1/b</math> is an integer
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| + <math>1/a - 1/b</math> is an integer
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| || Put <math>u = x^a</math>. Then,we get <math>x = u^{1/a}</math> and the integral becomes <math>\int e^{x^a} \, dx= \frac{1}{a} \int e^u u^{1/a - 1} \, du</math>. If <matH>1/a - 1/b</math> is an integer, then repeated use of integration by parts gets us to <math>\int e^u u^{1/b - 1} \, du</math>. Now, we plug back <math>y = u^{1/b}</math> and get <math>\int e^{y^b} \, dy</math>. Constants are ignored here as they don't affect the equivalence of integration problems.
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| - <matH>1/(ab)</math> is an integer
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| - <math>a/b</math> is an integer
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| </quiz> | | </quiz> |