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

Relationship integration math problem

  1. Jul 9, 2014 #1
    This is about attempting to solve ##\left( y'\right)^2 = y^2 - 1 ##.

    [tex]\int\frac{dy}{\sqrt{y^2 -1}} = \pm \int dx[/tex]

    using a trig. substitution and another trick,

    [tex]\int\frac{dy}{\sqrt{y^2 -1}} = \ln\left(y \pm \sqrt{y^2 - 1} \right) + C[/tex]

    I'm not sure about that [itex]\pm[/itex] sign. It came in when doing [itex]\tan (\sec^{-1} y)[/itex] since I had [tex]\sec \theta = y \longrightarrow\tan^2 \theta = y^2 -1\longrightarrow \tan \theta = \pm \sqrt{y^2 -1}[/tex]
    The solution is supposed to be something like [itex]y = \cosh x[/itex]. But I am not sure how I'm supposed to be getting that from separating and integrating.

    [itex]\ln\left(y + \sqrt{y^2 - 1} \right) = \pm x + C [/itex]
    [itex]y + \sqrt{y^2 - 1} = C_1e^{\pm x} [/itex]
  2. jcsd
  3. Jul 9, 2014 #2
    Hello Misterx

    Please see the attached picture. Not sure if I am correct. I suggest waiting for someone to verify things :smile:

    Attached Files:

  4. Jul 9, 2014 #3


    User Avatar
    Homework Helper

    You can do [tex]
    \int \frac{1}{\sqrt{y^2 - 1}}\,dy
    [/tex] by substituting [itex]y = \cosh u[/itex] so that [itex]dy = \sinh u\,du[/itex] and using the relationship [tex]\cosh^2 u - \sinh^2 u = 1.[/tex]
    Alternatively, if you solve [tex]
    y = \cosh x = \frac{e^{x} + e^{-x}}{2}
    [/tex] for [itex]x = \mathrm{arccosh}(y) \geq 0[/itex] you will find that [tex]\mathrm{arccosh}(y) = \ln\left(y + \sqrt{y^2 - 1}\right).[/tex]

    Note that substituting [itex]y = \cosh u[/itex] gets you to the result with considerably less effort than substituting [itex]y = \sec \theta[/itex].
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Relationship integration math problem
  1. Integration problem (Replies: 4)

  2. Integration problem (Replies: 3)

  3. Integral problem (Replies: 2)