Answer
$C=x \cos (xy)$
Work Step by Step
We are given
$[\cos(xy)−xy \sin(xy)]dx−x^2 \sin(xy)dy=0$
$\cos(xy) dx−xy \sin(xy)dx−x^2 \sin(xy)dy=0$
$\cos(xy) dx - [xy \sin(xy)dx + x^2 \sin(xy)dy]=0$
Consider:
$d(x\cos (xy))=\cos(xy) \times 1dx - [xy \sin(xy)dx + x \times x \sin(xy)dy]$
$\rightarrow d(x\cos (xy))=\cos(xy)dx - [xy \sin(xy)dx + x ^2\sin(xy)dy]$
Therefore: $d(x\cos (xy))=0$
$\int x\cos (xy)=\int 0$
$C=x \cos (xy)$
