Homework Help: How to solve integral

1. Jul 25, 2015

LagrangeEuler

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

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. Jul 25, 2015

micromass

Start by eliminating the fractions under the square root.

3. Jul 26, 2015

LagrangeEuler

Ok
$$C_1-K\cos y-\frac{C_2}{\sin^2 y}=\frac{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}{\sin^2 y}$$
So now I have
$$\int \frac{\sin ydy}{\sqrt{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}}$$

Last edited: Jul 26, 2015
4. Jul 26, 2015

LagrangeEuler

If I take
$\cos y =t$
then
$-\sin ydy=dt$
$$I=-\int\frac{dt}{\sqrt{C_1(1-t^2)-Kt(1-t^2)-C_2}}$$
$$I=-\int \frac{dt}{\sqrt{Kt^3-C_1t^2-Kt+C_1-C_2}}$$

5. Jul 28, 2015

geoffrey159

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.

6. Jul 28, 2015

Ray Vickson

You can express the integral in terms of Elliptic functions but it is very messy.

7. Jul 29, 2015

RUber

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.