$$x\sin(y)\,dy+(x^3-2x^2\cos(y)+\cos(y))\,dx$$ i tried solving the above d.e. the integrating factor comes out to be $e^{\int p\, dx}$ where $p$ was found out to be $x^2e^{-x^2}$.
But the resulting d.e after multiplying the integration factor turn out to be inexact. I can't figure out why.
$$x\sin(y)dy+(x^3-2(x^2)\cos(y)+\cos(y))dx=0$$ $$-xd\cos(y)+x^3dx+(-2x^2+1)\cos(y)dx=0$$ $$d\cos(y)-x^2dx+(2x-\dfrac 1 x)\cos(y)dx=0$$ The integrating factor should be : $$\mu(x)=e^{x^2-\ln x}=\dfrac {e^{x^2}}{x}$$ Then the DE becomes exact. $$d \left (\cos(y)\dfrac {e^{x^2}}{x} \right)-xe^{x^2}dx=0$$ $$d \left (\cos(y)\dfrac {e^{x^2}}{x} \right)-\dfrac 12d (e^{x^2})=0$$ Integrate.
It seems to me that your integrating factor should be: $$\mu(x)=-\dfrac {e^{x^2}}{x^2}$$ And not $x^2e^{-x^2}$