Answer
$9 \ln (2)$
Work Step by Step
Since, $\nabla f =\dfrac{\partial f}{ \partial x} i+\dfrac{\partial f}{ \partial y}j +\dfrac{\partial f}{ \partial z} k ...(1)$
We have: $ \dfrac{\partial f}{ \partial x}= 3x^2 ; \dfrac{\partial f}{ \partial y}=\dfrac{z^2}{y} \\ \dfrac{\partial f}{ \partial z}=2z \ln y$
Now, $f=x^3+g(y,z) ...(2)$
$\implies \dfrac{\partial g(y,z)}{ \partial x}=z^2 \ln y$
Equation (2) becomes: $f=x^3 +z^2 \ln y +h(z) .....(3)$
and $h(z)=C$
Substitute all the above values in the given integral as follows:
$ \int_{1,1,1}^{1,2,3} 3x^2 dx +\dfrac{z^2 dy}{y}+2z \ln y dz=1+9 \ln 2 -(1+0)= 9 \ln (2)$
