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July 25th, 2012, 10:43 AM   #1
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integrable functions

Is it possible to prove that for every integrable function there is only one solution(for example?(I dont mean with different constants, I mean is it possible to prove, that there is not any other function, which would solve that problem).
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July 25th, 2012, 11:15 AM   #2
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Re: integrable functions



Let's assume that we have two solutions , then



So if is the solution, another solution is only
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July 25th, 2012, 03:53 PM   #3
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Re: integrable functions

If i remember correctly some trigonometric functions have at least 2 different solutions for the integral. Can't remember which ones... was it or ?

I do remember being surprised about it and showing my calc professor years ago and he was speechless

Maybe someone else can elaborate more...
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July 25th, 2012, 03:55 PM   #4
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Re: integrable functions

I know there are cases where the anti-derivative may be expressed in different forms, but it can be shown these different forms are the same function.
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July 25th, 2012, 04:17 PM   #5
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Re: integrable functions

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Originally Posted by MarkFL
I know there are cases where the anti-derivative may be expressed in different forms, but it can be shown these different forms are the same function.
Maybe an example of what MarkFL is talking about is



and



But what i saw years ago was different and now i don't remember anything except i was genuinely surprised and professor didn't explain...
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July 25th, 2012, 04:28 PM   #6
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Re: integrable functions

Here is an example of multiple forms for the same integral, all differing by a constant:: http://en.wikipedia.org/wiki/Integral_o ... t_function
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July 25th, 2012, 04:38 PM   #7
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Re: integrable functions

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Originally Posted by The_Fool
Here is an example of multiple forms for the same integral, all differing by a constant:: http://en.wikipedia.org/wiki/Integral_o ... t_function
Yes, nice example, but are all 3 forms in the wiki article exactly the same function in disguise?
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July 25th, 2012, 05:51 PM   #8
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Re: integrable functions

I believe the cases I am thinking of derive from the following identities:

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July 25th, 2012, 07:07 PM   #9
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Re: integrable functions

Quote:
Originally Posted by agentredlum
Quote:
Originally Posted by The_Fool
Here is an example of multiple forms for the same integral, all differing by a constant:: http://en.wikipedia.org/wiki/Integral_o ... t_function
Yes, nice example, but are all 3 forms in the wiki article exactly the same function in disguise?
Correct. Another example is this integral:

To integrate this you'd need to do a trigonometric substitution. However, you could pick either sin or cos. If you work it out twice, one for each, you'll arrive at what appears to be different integrals. However, they are actually the same as they only differ by a constant.
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July 26th, 2012, 02:48 AM   #10
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Re: integrable functions

Since the differentiation of any function is unique; integration of any function is also unique.

If

Then it is possible to show that is a disguised .
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