I have to calculate the integral $$\oint_\Gamma{}{}xdx+(x+y)dy+(x+y+z)dz $$ where $\Gamma: x= \sin{t}, y=\cos{t}, z=\sin{t}+\cos{t}$ for $t\in[0, 2\pi]$.
I've applied the Stokes' theorem and got $$\iint_\Sigma{}{}dydz-dxdz+dxdy$$ I don't really know where to go from this. Integrals of vector fields are very confusing to me because we haven't covered this part of material in class and I cannot get my head around some examples as the ones from the lectures were much more comprehensible. I would really appreciate any kind of help.
Hint
A parametrisation of $\Sigma$ is given by
$$\Sigma=\{(r\cos (t),r\sin (t),r\cos(t)+r\sin(t))\mid r\in [0,1], t\in [0,2\pi] \}.$$