Find the general integral of (y+sin…

Find the general integral of $\dfrac{y+\sin x\cos^2(xy)}{\cos^2(xy)}dx+\dfrac{x}{\cos^2(xy)}dy+\sin y\,dy=0$.

Step 1: Split the equation

Rewrite as $\dfrac{y}{\cos^2(xy)}dx+\dfrac{x}{\cos^2(xy)}dy+\sin x\,dx+\sin y\,dy=0$, i.e. $\sec^2(xy)(y\,dx+x\,dy)+\sin x\,dx+\sin y\,dy=0$.

Step 2: Recognize exact differentials

Since $d(xy)=y\,dx+x\,dy$ and $\dfrac{d}{du}\tan u=\sec^2u$: $\sec^2(xy)\,d(xy)=d[\tan(xy)]$. Also $\sin x\,dx=d[-\cos x]$ and $\sin y\,dy=d[-\cos y]$.

Step 3: Sum the differentials

The equation is $d\big[\tan(xy)-\cos x-\cos y\big]=0$.