1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Integration using trig subs having trouble switching back to the original variable

  1. Oct 4, 2004 #1
    Hi, I carried out the integration until the very end...I don't know how to convert the variable back to the original one. :confused:

    [tex]\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{a^2-u^2}} [/tex]

    [tex] Let u = a\sin{\Theta}[/tex]
    [tex] du = a\cos{\Theta}d\Theta[/tex]

    The integral becomes...


    [tex]\csc{\Theta}d{\Theta} = \ln {|\csc{\Theta}-\cot{\Theta}|}+C[/tex]

    This is where I'm stuck. I don't know how to convert the thetas back into the "u"s. I haven't multiplied the answer by 1/a yet. I know that [tex]\Theta=\sin^{-1}{u/a}[/tex], but if I plug the [tex]\sin^{-1}{u/a}[/tex] into Theta, the expression becomes super messy and I really don't know what to do with it.

    Please help, thanks in advance! :smile:
    Last edited: Oct 4, 2004
  2. jcsd
  3. Oct 4, 2004 #2
    make a triangle.. it's the only way. For example, if theta = arcsin(u/a) then sin(arcsin(u/a)) = u/a, cos(arcsin(u/a)) = sqrt(a^2-u^2)/a. Make a triangle with sides u, sqrt(a^2-u^2) and a. You can find all the trig functions from it (be sure to label theta)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook