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?


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.