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November 9th, 2012, 06:53 PM   #1
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here comes the 1st challenging integral

I posted this challenging integral on another forum two years ago. It captured a lot of people's attention and discussion. They came up with different ways of analyzing it. I hope that it will spark the same level of interest on this forum.

Here is the challenging integral.

evaluate where x is any real number

What makes this integral challenging is that there is no specific function given to f(x). Can someone attempt this question for personal interest?
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November 9th, 2012, 08:22 PM   #2
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Re: here comes the 1st challenging integral

Where you have , is this supposed to be ?
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November 9th, 2012, 08:44 PM   #3
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Re: here comes the 1st challenging integral

The integrand is some function f(x+1/x) multiplied by (Inx) / x .

To clarify again, (Inx) is divided by x.
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November 9th, 2012, 08:54 PM   #4
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Re: here comes the 1st challenging integral

Is f differentiable for every x .
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November 9th, 2012, 08:56 PM   #5
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Re: here comes the 1st challenging integral

So, you don't mean the natural log of x but some product of the constants I and n times x over x?
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November 9th, 2012, 09:03 PM   #6
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Re: here comes the 1st challenging integral

Then the x's would cancel, surely he means natural logarihhm.
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November 9th, 2012, 09:08 PM   #7
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Re: here comes the 1st challenging integral

That is my thought too, but 3 times now the character I (uppercase i) has been used instead of l (lowercase L), and while they look similar, surely one knows the l stands for logarithm. I just want it to be clear what the integrand is. Not trying to be "picky."
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November 9th, 2012, 09:45 PM   #8
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Re: here comes the 1st challenging integral

Yes, it is the natural logarithm of x.

In this problem, there is no information about whether the function is differentiable for all x.

What you see is all the information given in the problem.
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November 9th, 2012, 10:00 PM   #9
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Re: here comes the 1st challenging integral

I think I have managed to solve it , waiting for
other solutions to arise .
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November 9th, 2012, 10:08 PM   #10
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Re: here comes the 1st challenging integral

Is it of the form recursive to it's derivative (with the integration by parts terms in too)?
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