Let $F=\left[\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right].$ Let $C$ by the curve consisting of the line segments from $$(-1,0)\to (0,-2)\to (2,0)\to (3,4)\to (0,5)\to (-1,0)$$ Find $\int_{C}{\bf{F}}\cdot d{\bf{s}}.$
i think the answer is 0 because F is conservative, but how can i find the integration limits at each of the continuous segments of straight line.
Your line integral may be written as
$$\oint_C P(x,y) dx + Q(x,y) dy$$
where $P(x,y) = x/(x^2+y^2)$ and $Q(x,y) = y/(x^2+y^2)$. The line integral is evaluated by parametrizing each line segment and integrating over the parameter. I will do one of the line segments as an example - you should be able to do the rest on your own.
Consider the line segment between $(-1,0)$ and $(0,-2)$. The equation of the line is $y=-2 x-2$. Ths, let $x=t$, $y=-2 t-2$. Then $dx=dt$, $dy=-2 dt$. Note that on this line segment, $t \in [-1,0]$. (Note that $t$ just follows $x$. That's how you determine the limits on each segment.)
The integral over this segment is
$$\int_{-1}^0 dt \left (\frac{t}{t^2+4 (t+1)^2} - 2 \frac{-2 (t+1)}{t^2+4 (t+1)^2}\right ) = \int_{-1}^0 dt \frac{5 t+4}{5 t^2+8 t+4}$$
which you can do out yourself easily and find is equal to $\log{2}$.
Now repeat for the other four line segments and add up the results.