Let H be the segment of the hyperbola x-y°= | Connecting (1,0) to (2, J3). Evaluate the line, integral %3D Suly dx t x dy

Given 

H be the segment of hyperbola $x^2-y^2=1$ connecting the points (1,0) to (2,$\sqrt{3}$) 

we have to evaluate the integral

${\int }_H\left(y+\frac{1}{x}\right)dx+xdy$

Now by using Green's theorem we know that

${\int }_{c}Mdx+Ndy =\underset{s}{\iint }\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy$