I have been working in this problem about line integrals.
Compute $\int_C \mathbf{f} d\mathbf{r}$ where $\mathbf{f}(x,y,z)=(2x,\cos(y)\cos(z),\sin(y)sin(z))$ and $C$ is the polygonal which links $(0,0,0),(0,0,1),(0,1,1),(1,1,1),(1,1,0),(1,0,1) in this order.
The first thing that came to my mind was using Poincaré lemma to verify if $f$ was conservative: $D_1f_2=0=D_2f_1$, $D_1f_3=0=D_3f_1$ but $D_3f_2=-\cos(y)\sin(z)\neq \cos(y)\sin(z)=D_2f_3$.
I did this so I could use the path independence and reduce the amount of work. But as $f$ seems to not be conservative I assume I just have to parametrize every segment and do the calculus but I have the feeling that I may be missing something since it is "almost" (just by a sign) conservative. The only thing that came to my mind was splitting the original field into $(2x,\cos(y)cos(z),-\sin(y)\sin(z))+(0,0,2\sin(y)\sin(z))$ so the first one is conservative but that does not reduce it too much.
Is there any "trick" on how to proceed in this case or do I have to do it the long way?