Incomplete elliptic integral and Jacobi's form The incomplete elliptic integral of the first kind is written (using trigonometric form) : $ F(\varphi,k)=\int_{0}^{\varphi} \frac{1}{\sqrt{1-k^2 \sin^2(\theta)}} \mathrm{d}\theta $.
Then, it is noted everywhere that if we make the change of variable $t=\sin(\theta)$, then the integral can be re-written in the so-called Jacobi's form : $F(x,k)=\int_{0}^{x} \frac{1}{\sqrt{ (1-t^2)(1-k^2 t^2) }} \mathrm{d}t$, where $x$ is used instead of $\sin(\varphi)$...
So good up to there, but, $t=\sin(\theta) \Longrightarrow \mathrm{d}t=\cos(\theta)\mathrm{d}\theta$, and, depending on $\theta$, $\cos(\theta)=\pm\sqrt{1-\sin^2(\theta)}=\pm\sqrt{1-t^2}$
So I wonder why, in the form of Jacobi, we use the positive writing of $\cos(\theta)$ ? 

Because the original integral makes sense for $\theta$ in a neighborhood of $0$. There, $\cos$ is positive.
