0
$\begingroup$

I want to try and find an expression for the integral \begin{equation} \label{eq1} \int_a^b\frac{dx}{\sqrt{c_1+c_2x+c_3x^2+c_4x^3+c_5x^4}}\hspace{2cm}(1) \end{equation} I found this paper 'Nonlinear dynamics in periodic phase space' by A. Iomin, D. Gangardt, and S. Fishman where they deal with an expression of the form $$I=\int_0^q\frac{dq'}{\sqrt{1-(H_0-\cos(q'))^2}}\hspace{2cm}(2)$$ where they changed variables and reduced with a change of variables ($x'=\cos(q')$) to a form like (1) where $a=x$ and $b=1$ and the answer was of the form of an elliptic integral. So:

$$I=F(\psi,k)\hspace{2cm}(3)$$ where $$\psi=\arcsin\left(\sqrt{\frac{2(1-\cos q}{(2+H_0)(1-H_0-\cos q)}}\right)$$ and $$k=\sqrt{1-\frac{H_0^2}{4}}.$$ I want an expression of the form (3), but for general limits. I've looked at integral tables and mathematica but I've had no luck. Can anybody point me to a formula that could help?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.