Integrate $\int_{C}{\frac{-x}{x^2+y^2}dx+\frac{y}{x^2+y^2}dy}$
C: $x=cost$, $y=sint$, $\quad0\le t\le \frac{\pi}{2}$
I know I can get answer easily using $x^2+y^2=1$.
What I'm confused is when I integrate it not considering C.
It leads to $\frac{-1}{2}\ln{(x^2+y^2)}|_{a}^{b}+\frac{1}{2}\ln{(x^2+y^2)}|_{c}^{d}$
And whatever a,b,c, and d, it equals to $0$ which is not a correct answer.
It seems has to do with parametrization but I can't quite pick what went wrong.
Could you tell me why it doesn't give me an answer which is 1?
You may not treat $y$ as constant for the left integral, and similarly on the right. So
$$\int_C\frac{x}{x^2+y^2}dx\ne\frac12\log(x^2+y^2).$$
In fact you have
$$\int_C\frac{x}{x^2+y^2}dx=\int_a^b\frac{x}{x^2+y^2(x)}dx=\int_a^b\frac x{x^2+1-x^2}\,dx.$$