all values of line integral

Question

Find all values of the line integral $\int_{(1,0)}^{(2, 2)}{\frac{-y}{x^2 + y^2}dx + \frac{x}{x^2 + y^2}dy}$, by a path that does not contains the point $(0,0)$

Answer 1

hint

For the path, take the line whose equation is $$y=\frac{2-0}{2-1}(x-1)=2x-2$$ So, $ dy=2dx$. The integrale becomes $$\int_1^2\frac{(2x-2x+2)dx}{4x^2-8x+4+x^2}$$