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How to solve integral

  1. Jul 25, 2015 #1
    1. The problem statement, all variables and given/known data
    How to solve integral
    [tex]\int \frac{dy}{\sqrt{C_1-K \cos y-\frac{C_2}{\sin^2 y}}}[/tex]

    2. Relevant equations
    ##C_1,C_2## and ##K## are constants.

    3. The attempt at a solution
    I am not sure which method I should use here or is this integral maybe eliptic? Please give me the hint. Which supstitution or method and I will solve integral to the end.
     
  2. jcsd
  3. Jul 25, 2015 #2

    micromass

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    Start by eliminating the fractions under the square root.
     
  4. Jul 26, 2015 #3
    Ok
    [tex]C_1-K\cos y-\frac{C_2}{\sin^2 y}=\frac{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}{\sin^2 y}[/tex]
    So now I have
    [tex]\int \frac{\sin ydy}{\sqrt{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}} [/tex]
     
    Last edited: Jul 26, 2015
  5. Jul 26, 2015 #4
    If I take
    ##\cos y =t##
    then
    ##-\sin ydy=dt##
    [tex]I=-\int\frac{dt}{\sqrt{C_1(1-t^2)-Kt(1-t^2)-C_2}}[/tex]
    [tex]I=-\int \frac{dt}{\sqrt{Kt^3-C_1t^2-Kt+C_1-C_2}} [/tex]
     
  6. Jul 28, 2015 #5
    If your polynomial at the denominator were of degree 2, you would write it under its canonical form, and depending upon the coefficients and the discriminant, you would use a substitution of type Arcsin, Argsinh, or Argcosh to get rid of the square root. I don't know if you can do the same thing for polynomials of degree 3.
     
  7. Jul 28, 2015 #6

    Ray Vickson

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    You can express the integral in terms of Elliptic functions but it is very messy.
     
  8. Jul 29, 2015 #7

    RUber

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    Are there limits on your integration?
    It looks like the function of y might be defined for y in (0,pi)+k*pi for integer values of k, and may be imaginary for certain choices of constants.
     
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