July 25th, 2012, 10:43 AM  #1 
Member Joined: Apr 2010 Posts: 43 Thanks: 0  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).

July 25th, 2012, 11:15 AM  #2 
Senior Member Joined: Jul 2011 Posts: 118 Thanks: 0  Re: integrable functions Let's assume that we have two solutions , then So if is the solution, another solution is only 
July 25th, 2012, 03:53 PM  #3 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  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... 
July 25th, 2012, 03:55 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs  Re: integrable functions
I know there are cases where the antiderivative may be expressed in different forms, but it can be shown these different forms are the same function.

July 25th, 2012, 04:17 PM  #5  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: integrable functions Quote:
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...  
July 25th, 2012, 04:28 PM  #6 
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  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 
July 25th, 2012, 04:38 PM  #7  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: integrable functions Quote:
 
July 25th, 2012, 05:51 PM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 466 Math Focus: Calculus/ODEs  Re: integrable functions
I believe the cases I am thinking of derive from the following identities: 
July 25th, 2012, 07:07 PM  #9  
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  Re: integrable functions Quote:
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.  
July 26th, 2012, 02:48 AM  #10 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  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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