Answer
$e+15 $
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 $F(x,y,z)=(e^{x+y}, e^{x+y}, e^{x+y} z)$
So, we can see that $\dfrac{\partial f_1}{\partial y}=e^{x+y}=\dfrac{\partial f_2}{\partial x}$ and $\dfrac{\partial f_2}{\partial z}=0=\dfrac{\partial f_3}{\partial y}$ and $\dfrac{\partial f_1}{\partial z}=0=\dfrac{\partial f_3}{\partial x}$
This means that the given vector field is not conservative.
So, we can expressed the integral as:
$\oint F \ dr=\int_0^{1} F[r(t)] r'(t) \ dt\\=\int_0^{1} (e^t, e^t , -4t e^t) (-1, 2, -4) \ dt \\=\int_0^{1} (16t +1) \times e^t \ dt \\=[16(1) -15] e^1-[16(0) -15] e^0 \\=e+15 $
