If F(x, y, z) = i + sin(z) j + y cos(z) k is conservative, then there exists a scalar function f(x, y, z) such that grad(f) = F, which means
∂f/∂x = 1
∂f/∂y = sin(z)
∂f/∂z = y cos(z)
Integrating each each of these equations gives
∫ ∂f/∂x dx = ∫ dx ⇒ f(x, y, z) = x + α(y, z)
∫ ∂f/∂y dy = ∫ sin(z) dy ⇒ f(x, y, z) = y sin(z) + β(x, z)
∫ ∂f/∂z dx = ∫ y cos(z) dz ⇒ f(x, y, z) = y sin(z) + γ(x, y)
It follows that α(y, z) = y sin(z) and β(x, z) + γ(x, y) = x + C where C is a constant. So
f(x, y, z) = x + y sin(z) + C
and F is indeed conservative.
