Advanced Integration techniques Needed

[MBM1]

Hi I have the integral

$G(x,y)=\int_{0}^{2\pi}d\theta\frac{1}{\sqrt{a_1\cos(2\theta)+a_2\sin(2\theta)+1}}\frac{1}{x\cos(\theta)+y\sin(\theta)}$I broke it into two terms in the hopes of simplifying the integrand,

$G(x,y)=\int_{0}^{2\pi}d\theta(\frac{1}{a_1\cos(2\theta)+a_2\sin(2\theta)}-\frac{\sqrt{a_1\cos(2\theta)+a_2\sin(2\theta)+1}}{a_1\cos(2\theta)+a_2\sin(2\theta)})\frac{1}{x\cos(\theta)+y\sin(\theta)}$

By posing restrictions on the values of $a_1$,$a_2$,$x$,$y$ I could represent the sum of the cofunctions to one term.

for $a_1>0$,$a_2$ any number between $-\infty$ and $+\infty$

$a_1\cos(2\theta)+a_2\sin(2\theta)= A_m\cos(2\theta-2\theta_m)$

for $a_1<0$,$a_2$ any number between -$\infty$ and +$\infty$

$a_1\cos(2\theta)+a_2\sin(2\theta)= -A_m\cos(2\theta-2\theta_m)$

for $x>0$,$-\infty<y<\infty$

$x\cos(\theta)+y\sin(\theta)= B_m\cos(\theta-\theta_{xy})$

for $x<0$,$-\infty<y<\infty$

$x\cos(\theta)+y\sin(\theta)= -B_m\cos(\theta-\theta_{xy})$

Therefore for $a_1>0$,$a_2$ any number between -$\infty$ and +$\infty$, $x>0$ and $-\infty<y<\infty$
$G(x,y)= \int_{0}^{2\pi}d\theta (\frac{1}{A_m\cos(2\theta-2\theta_m)}\frac{1}{B_m\cos(\theta-\theta_{xy})}-\frac{\sqrt{1+A_m\cos(2\theta-2\theta_m)}}{A_m\cos(2\theta-2\theta_m)} \frac{1}{B_m\cos(\theta-\theta_{xy})})$

I can solve the term on the left however I am finding difficulties to solve the one on the right. Any help would be appreciated.

 

[jvicens]

Hello,

Thank you for your post. It seems like you have made some progress in simplifying the integrand by breaking it into two terms and using restrictions on the values of the parameters. However, as you mentioned, you are having difficulties solving the term on the right.

One suggestion would be to try using trigonometric identities to simplify the term on the right. For example, you could use the double angle formula for cosine, $\cos(2\theta) = 2\cos^2(\theta) - 1$, to rewrite the term as $\frac{\sqrt{1+2A_m\cos^2(2\theta-2\theta_m)}}{A_m\cos(2\theta-2\theta_m)} \frac{1}{B_m\cos(\theta-\theta_{xy})}$. Then, you could use the Pythagorean identity, $\sin^2(\theta) + \cos^2(\theta) = 1$, to simplify the square root.

Another approach would be to use a substitution to transform the integral into a more manageable form. For example, you could try substituting $u = \cos(\theta)$, which would change the limits of integration to $-1$ and $1$ and would make the integrand a rational function.

I hope these suggestions are helpful. Keep exploring and trying different approaches, and you will eventually find a solution. Good luck!