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  #1  
Old 11-02-2006, 01:15 PM
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Default Integration by parts shortcut

I need to integrate x^4*e^(-2x)....any suggestions?
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Old 11-02-2006, 01:31 PM
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Quote:
Originally Posted by GraffixMan View Post
I need to integrate x^4*e^(-2x)....any suggestions?
Tabular integration by parts. Here's how it works: Write down the derivatives of alongside the antiderivatives of :













Now form the alternating sum of the products of the k(th) entry in the first column with the (k+1)th entry in the second column until such products are not possible:



which in turn simplifies to

.

This is the required antiderivative.
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Old 11-02-2006, 01:32 PM
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What he said.
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Old 11-02-2006, 01:37 PM
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Neato. Good tip.
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Old 11-02-2006, 01:42 PM
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This should be a sticky! I seriously have never heard this trick...ever.

Does it work with anything (i.e. instead of exponentials, would it work with trig functions - as long as one terminations with increasing derivatives?)
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Old 11-02-2006, 02:03 PM
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Tabular integration is sometimes taught at the AP Calculus BC level, and is mentioned in the Wikipedia entry on integration by parts. It is merely a bookkeeping method for a recursive integration. (Bookkeeping, by the way, is one of the few words which have three consecutive double letter pairs.) Thus the method is valid for any integrand that is the product of two functions, one of which has an n(th) derivative that is zero, and the other has n successive antiderivatives.

Incidentally, this is a method by which one can easily prove the formula for the CDF of a gamma distribution.
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Old 11-02-2006, 02:12 PM
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Quote:
Originally Posted by atomic View Post
Tabular integration is sometimes taught at the AP Calculus BC level, and is mentioned in the Wikipedia entry on integration by parts. It is merely a bookkeeping method for a recursive integration. (Bookkeeping, by the way, is one of the few words which have three consecutive double letter pairs.) Thus the method is valid for any integrand that is the product of two functions, one of which has an n(th) derivative that is zero, and the other has n successive antiderivatives.

Incidentally, this is a method by which one can easily prove the formula for the CDF of a gamma distribution.
IIRC (I'd need to do it again), a buddy and I both (independently) solved a problem he had in a "book of problems" similarly. However, both multiples were recursive (one was trig and the other exponential IIRC). I'll have to ask him for the problem again, but I believe he intended to generalize it and translate it into a useful Laplace transform.

(it's probably been done before, but we're undergrads, and it's good practice ).
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Old 11-02-2006, 02:25 PM
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Doesn't Laplace have bounds from 0 to infnity so lots of things cancel?
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Old 11-02-2006, 03:15 PM
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tic-tac-toe method
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Old 11-02-2006, 03:41 PM
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Isnt that tabular method just a more organized way of doing nested uv-int(vdu) over and over again until du is 1?
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