Let

$$\begin{aligned} u &=\left(2 x+x^{2} z+z^{3}\right) \exp ((x+y) z) \\ v &=\left(x^{2} z+z^{3}\right) \exp ((x+y) z) \\ w &=\left(2 z+x^{3}+x^{2} y+x z^{2}+y z^{2}\right) \exp ((x+y) z) \end{aligned}$$

Show that $u d x+v d y+w d z$ is an exact differential, clearly stating any criteria that you use.

Show that for any path between $(-1,0,1)$ and $(1,0,1)$

$$\int_{(-1,0,1)}^{(1,0,1)}(u d x+v d y+w d z)=4 \sinh 1$$