How to calculate $$ \int_{0}^{2K(k)} dn(u,k)^2\;du?$$ Where $dn$ is the Jacobi Elliptical function dnoidal and $k \in (0,1)$ is the modulus. I know from the Fórmula $(110.07)$ of [1] (see page 10) that $$ \int_{0}^{K(k)} dn(u,k)^2\;du=E(k),$$ where $E$ is the normal elliptic integral of the second kind complete. For this I can conclude that $$ \int_{0}^{2K(k)} dn(u,k)^2\;du=2E(k)?$$
[1] P. F. Byrd. M. D. Friedman. Hand Book of Elliptical Integrals for Engineers and Scientis. Springer-Verlag New York Heidelberg Berlim, $1971$.
I don't know much about these functions but I believe that it is not necessarily true since: $$I=\int_0^{2K(k)}dn(u,k)^2du$$ $v=u/2$ then $du=2dv$ and our integral becomes: $$I=2\int_0^{K(k)}dn(2v,k)^2dv$$ and this term of $2v$ rather than $v$ could change the value of the integral depending on what this $dn$ function is