1
$\begingroup$

I want to know if $\displaystyle{\int_{0}^{+\infty}\frac{e^{-x} - e^{-2x}}{x}dx}$ is finite, or in the other words, if the function $\displaystyle{\frac{e^{-x} - e^{-2x}}{x}}$ is integrable in the neighborhood of zero.

  • 0
    the function that I mean is integrable or not integrable?2012-12-13
  • 0
    It seems to me that it should be. Our primary concern is that the function might "blow up" around $x=0$. However, we can check the value of the integrand at zero. By L'Hopital's rule, $\lim_{x \to 0} \frac{e^{-x}-e^{-2x}}{x} = \lim_{x \to 0} \frac{-1+2}{1}$. So, the function seems to behave well around zero, so I suspect it will be integrable on a neighborhood about zero. Also, I checked on Wolfram Alpha and it says that it is integrable.2012-12-13
  • 0
    how I can calculate it?2012-12-13
  • 1
    It equals $\ln 2$: http://www.wolframalpha.com/input/?i=integrate+%28e%5E%28-x%29-e%5E%28-2*x%29%29%2Fx+from+0+to+infinity2012-12-13
  • 0
    I haven't understand how I can find Ln22012-12-14

5 Answers 5