Answer
$$0$$
Work Step by Step
We are given that $\phi(x,y, z)=1+x^2 yz$
Also, the curve is $r(t)= (\cos (2t), \sin (2t), t)$
Therefore, the integral is:
$\int_C \nabla \phi \ dr=\phi[r(4 \pi)]-\phi[r(0)] \ dt\\=\phi (1, 0, 4 \pi) -\phi (1,0,0) \\=1+(1)^2(0)(4 \pi) -[1+(1)^2(0)(0)] \\=1-1\\=0 $
