Answer
$26$
Work Step by Step
When $F(x,y)=pi+qj$ is a conservative field, then throughout the domain $D$, we get
$\dfrac{\partial p}{\partial y}=\dfrac{\partial q}{\partial x}$
$p$ and $q$ are the first-order partial derivatives on the domain $D$.
Here, we have $f_x(x,y)=2x+y$ and $f_y(x,y)=x$
$f(x,y)=x^2+xy+g(y)$ [g(y) : A function of y]
$f_y(x,y)=x+g'(y)$
Here, $g(y)=k$
Thus, $f(x,y)=x^2+xy+k$
Now, $W=\int_C F \cdot dr =f(4,3)-f(1,1)=(16+12+k)-(1+1+k)=26$
