# 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.