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The integral to be calculated is: $$I=\int_{a}^{b}\frac{f(\frac{a}{x})-f(\frac{b}{x})}{x}dx$$

Really this is the only info given. There is nothing about the nature of $f(x)$ or $a,b$. Since I couldn't apply any of the properties of the definite integral, I thought that perhaps $f(x)$ is to be chosen to our convenience since there's nothing given.

So with some standard assumptions I put $f(x)=\ln x$ and got the value $I=1$ but the answer given is $I=0$. Now my question is whether my approach is correct or this really is a case of data insufficiency (even with some standard assumptions)? Or perhaps someone could provide an alternate approach?

Jose Avilez
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infinite-blank-
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  • With $f(x) = \ln(x)$ and $0 < a < b$ I get $I=-(\ln \frac ba)^2$ while with $f(x)=7$ I get $I=0$ – Henry Sep 20 '19 at 17:13
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    In the case that the bounds were meant to be $[0,\infty)$ instead of $[a,b]$, this would be Frullani's integral. That aside, you definitely need more information and the integral is certainly not zero in general. (And in the case of $f(x)=\ln(x)$, it is $-\ln^2(b/a)$, not one...) – Simply Beautiful Art Sep 20 '19 at 17:25
  • @infinite-blank- With $f(x)=\ln(x)$ the integrand is negative so the integral is too – Henry Sep 20 '19 at 17:28
  • @SimplyBeautifulArt Can you provide some links related to Frullani's Integral? Not too advanced though... – infinite-blank- Sep 20 '19 at 17:44
  • you can check here: https://math.stackexchange.com/questions/61828/proof-of-frullanis-theorem – ThomasL Sep 20 '19 at 18:46

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