Trouble with integral $\int 2y\,\text{d}x+\int 2x\,\text{d}y$ arising from a physics problem

Question

I was working on the below physics problem and I faced problem with a mathematical part of it:

A particle of mass $2$kg is subjected to a two dimensional conservative force given by $$F_x=-2x+2y,\;\;\;\;F_y=2x-y^2$$ (where $x$, $y$ is in metres and $F$ is in Newtons). If the particle has kinetic energy $\frac{8}{3}$ Joules at the point $(2,3)$, find the speed of the particle when it reaches the point $(1,2)$.

I encountered the integral $$\int 2y\,\text{d}x+\int 2x\,\text{d}y.$$ Mathematically this would be $2(2xy)$, but for getting the correct solution, I have to use only one $2xy$ and put limits. So what I am doing wrong?

https://www.doubtnut.com/question-answer-physics/-a-particle-of-mass-2kg-is-subjected-to-a-two-dimensional-conservative-force-given-byfx-2x-2yfy2x-y2-214111078

Edit: In comments and answer,many have given thier approach but I think my mai question has not been answer that 2xy term should come 2 times so shouldn't it be 4xy?

Answer 1

$F_x=-2x+2y$, $F_y=2x-y^2$

$W=\int\vec{F}\cdot d\vec{x}=(-2x+2y)dx+(2x-y^2)dy=-x^2+2xy-y^3/3|_{(2,3)}^{(1,2)}=?$

Alternatively, $\int F_x dx=-x^2+2xy+g(y)$.

$\frac{d}{dy}\int F_xdx=2x+g'=2x-y^2$

So $g(y)=-y^3/3+C$.

Let potential $V(x,y)= -x^2+2xy-y^3/3$ evaluate at the endpoinds then take the difference, get the same answer.