Find the general integral of ((x)/(sin y)+2)dx+((x^2+1)cos…

Find the general integral of $\left(\dfrac{x}{\sin y}+2\right)dx+\dfrac{(x^2+1)\cos y}{\cos2y-1}dy=0$.

Step 1: Simplify and check exactness

Since $\cos2y-1=-2\sin^2y$, $N=-\dfrac{(x^2+1)\cos y}{2\sin^2y}$. Then $M_y=N_x=-\dfrac{x\cos y}{\sin^2y}$, so the equation is exact.

Step 2: Find the potential function

$F=\int\!\left(\dfrac{x}{\sin y}+2\right)dx=\dfrac{x^2}{2\sin y}+2x+g(y)$. Matching $F_y=-\dfrac{x^2\cos y}{2\sin^2y}+g'(y)=N$ gives $g'(y)=-\dfrac{\cos y}{2\sin^2y}$, so $g(y)=\dfrac{1}{2\sin y}$.