Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Simple integral

  1. Jul 4, 2016 #1
    ##\int\sqrt{\frac{x}{x-a}}dx=?##
     
  2. jcsd
  3. Jul 4, 2016 #2

    QuantumQuest

    User Avatar
    Gold Member

    What have you tried so far?
     
  4. Jul 4, 2016 #3
  5. Jul 4, 2016 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    I don't get it. I clicked the wolfram link. What is so strange and ridiculous about the solution?
     
  6. Jul 4, 2016 #5

    QuantumQuest

    User Avatar
    Gold Member

    The differences are due to simplifications and / or different ways to write the same thing.
    Have you tried to do the integration? I give you the result if you want to try it out:

    ##\dfrac{a\left(\ln\left(\sqrt{x-a}+\sqrt{x}\right)-\ln\left(\left|\sqrt{x-a}-\sqrt{x}\right|\right)\right)}{2}+\sqrt{x}\sqrt{x-a}##
     
  7. Jul 4, 2016 #6
    I easily compute this integral - just make a simple substitute:
    ##x = a\cosh^2(t)##
    then:
    ##dx = 2a\cosh(t)\sinh(t)##
    and
    ##x-a = a\sinh^2(t)##
    so, the integral now is:
    ##\int cosh^2(t)dt = \int(\cosh(2t)+1)dt = \frac{1}{2}\sinh(2t)+t+C=\sinh(t)\cosh(t)+t+C##

    finally:
    ##I = \sqrt{x(x-a)}+a\cdot arcosh{\sqrt{x/a}}##
     
  8. Jul 4, 2016 #7

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You need to get back to the variable ##x##.
     
  9. Jul 4, 2016 #8
    So, I don't what is going on.

    The proposed solutions are quite nonsensical - what it the reason?

    ##arcosh(x)=\ln(x+\sqrt{x^2-1})##
     
  10. Jul 4, 2016 #9

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Can you tell us why they are nonsensical?
     
  11. Jul 4, 2016 #10
    I showed this already: #3.
    And there are much more idiotic versions in the net!
     
  12. Jul 4, 2016 #11

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    The links you posted in #3 are absolutely correct, so I have no idea what you're talking about.
     
  13. Jul 4, 2016 #12
    Try to compute some definite integral using these 'alternative solutions' then You get it.

    For example:
    what is a correct distance, means: according to the GR, to the Sun from the Earth?
     
  14. Jul 4, 2016 #13

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Can you please be more specific?
     
  15. Jul 4, 2016 #14
    The question: 'what is a distance...', and with a given metric is rather very precise - there is no room for any more specification.
     
  16. Jul 4, 2016 #15

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You're making no sense, sorry.
     
  17. Jul 6, 2016 #16
    OK, I sorry.
    The result is the same.
    ##2\ln(\sqrt{x}+\sqrt{x-1})=\ln(\sqrt{x}+\sqrt{x-1})^2=\ln(2\sqrt{x}\sqrt{x-1}+x+x-1)##

    But look at this.
    The distance with the Schwarzschild metric is equal to:
    ##s=\sqrt{r(r-a)} + a.arcosh(\sqrt{r/a})##
    thus for the case of very big distances: ##r >> a##, the distance is approximately:
    ##s\approx r + a\ln(4r/a)##

    thus to the Sun it is some bigger distance, than the simple: r = 150 mln km,
    because it's about: ##ds = a\ln(4r/a)## bigger.
    a = 3km for the Sun, so this is:
    ##ds = 3\ln(4*150mln / 3) = 36.6 km## more.
     
    Last edited: Jul 6, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Simple integral
  1. Simple Integration (Replies: 5)

  2. Simple integration (Replies: 2)

  3. Simple integral (Replies: 4)

  4. Simple Integration (Replies: 2)

Loading...