Let u=xsin(t)+tcos(x), then:
du/dt=(dx/dt)sin(t)+xcos(t)+cos(x)-tsin(x)(dx/dt)=
(sin(t)-tsin(x))(dx/dt)+xcos(t)+cos(x).
Therefore du=(cos(x)+xcos(t))dt+(sin(t)-tsin(x))dx=0 (from the given equation).
du/dt=(cos(x)+xcos(t))+(sin(t)-tsin(x))(dx/dt)=0.
But if du/dt=0, u=C a constant, therefore:
xsin(t)+tcos(x)=C is the solution.