Given $\frac{dy}{dx}=x+\sin x\cos y+x\cos y+\sin x$

$\Rightarrow \frac{dy}{dx}=x\left(1+\cos y\right)+\sin x\left(1+\cos y\right)$

$\Rightarrow \frac{dy}{1+\cos y}=\left(x+\sin x\right)dx$

Integrating both sides,

$\Rightarrow \int \frac{dy}{1+\cos y}=\int \left(x+\sin x\right)dx$

$\Rightarrow \int \frac{dy}{2{\cos }^2{\displaystyle \frac{y}{2}}}=\int \left(x+\sin x\right)dx \left[\therefore 1+\cos 2A=2{\cos }^{2}A\right]$

$\Rightarrow \frac{1}{2}\int {\sec }^2\frac{\mathrm{y}}{2}\mathrm{dy}=\int \left(x+\sin x\right)dx$

$\Rightarrow \tan \frac{y}{2}=\frac{x^2}{2}-\cos x+C$