Answer
$25$
Work Step by Step
For a vector field to be Conservative, $\dfrac{\partial f_1}{\partial y}=\dfrac{\partial f_2}{\partial x}$ and $\dfrac{\partial f_2}{\partial z}=\dfrac{\partial f_3}{\partial y}$ and $\dfrac{\partial f_1}{\partial z}=\dfrac{\partial f_3}{\partial x}$
We are given that the force field as: $F(x,y, z)=(x,y,z)$
So, we can see that $\dfrac{\partial f_1}{\partial y}=0=\dfrac{\partial f_2}{\partial x}$ and $\dfrac{\partial f_2}{\partial z}=0=\dfrac{\partial f_3}{\partial y}$ and $\dfrac{\partial f_1}{\partial z}=0=\dfrac{\partial f_3}{\partial x}$
Next, we will find the potential function for the given vector field as:
$\phi(x,y,z)=\dfrac{x^2+y^2+z^2}{2}$
Therefore, the integral can be expressed as:
$\int_C F \ dr=\phi(2, 4, 6) -\phi(1, 2, 1) \\=\dfrac{(2)^2+(4)^2+(6)^2}{2}-[\dfrac{(1)^2+(2)^2+(1)^2}{2}]\\ =25$
