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 November 9th, 2012, 06:53 PM #1 Senior Member   Joined: Apr 2008 Posts: 194 Thanks: 3 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 $\int_0^{\infty} f( x + \frac{1}{x} ) \frac{Inx}{x} dx$ 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?
 November 9th, 2012, 08:22 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: here comes the 1st challenging integral Where you have $Inx$, is this supposed to be $\ln(x)$ ?
 November 9th, 2012, 08:44 PM #3 Senior Member   Joined: Apr 2008 Posts: 194 Thanks: 3 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.
 November 9th, 2012, 08:54 PM #4 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions Re: here comes the 1st challenging integral Is f differentiable for every x .
 November 9th, 2012, 08:56 PM #5 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 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?
 November 9th, 2012, 09:03 PM #6 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions Re: here comes the 1st challenging integral Then the x's would cancel, surely he means natural logarihhm.
 November 9th, 2012, 09:08 PM #7 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs 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."
 November 9th, 2012, 09:45 PM #8 Senior Member   Joined: Apr 2008 Posts: 194 Thanks: 3 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.
 November 9th, 2012, 10:00 PM #9 Math Team     Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions Re: here comes the 1st challenging integral I think I have managed to solve it , waiting for other solutions to arise .
 November 9th, 2012, 10:08 PM #10 Senior Member     Joined: Jul 2012 From: DFW Area Posts: 642 Thanks: 99 Math Focus: Electrical Engineering Applications 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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