so I wanted to compare my computations in order to know whether they're flawless:
Given the vector field $v_2(x_1,x_2,x_3)=\frac{1}{2}\begin{pmatrix} 2x_1x_2x_3\\x_1^2x_3\\x_1^2x_2 \end{pmatrix}$ and using line integration, integrating along $K: \begin{pmatrix} 0\\0\\0 \end{pmatrix}\overset{K_1}\longrightarrow \begin{pmatrix} x\\0\\0 \end{pmatrix}\overset{K_2}\longrightarrow \begin{pmatrix} x\\y\\0 \end{pmatrix}\overset{K_3}\longrightarrow \begin{pmatrix} x\\y\\z \end{pmatrix}$, I found: $$f(x,y,z)=\int_{0}^{x}f_x(t,0,0) \, dt+\int_{0}^{y}f_y(x,t,0) \, dt+\int_{0}^{z}f_z(x,y,t) \, dt=\int_{0}^{x}0 \, dt+\int_{0}^{y} \, 0 \, dt+\int_{0}^{z}x^2y \, dt=x^2yz+C $$
by solving $$\frac{1}{2}\begin{pmatrix} 2x_1x_2x_3\\x_1^2x_3\\x_1^2x_2 \end{pmatrix}\overset{!}=\frac{1}{2}\begin{pmatrix} f_x\\f_y\\f_z \end{pmatrix}=\nabla f$$
When deriving $f(x,y,z)$ it's correct, but is the computation fine?