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Multivariable Calculus

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Difficult

Second Type Line Integral Along Ellipse

MVCALC-XXVXRZ

Evaluate the line integral:

$$I=\int_C \frac{y}{x^2+y^2}dx$$

...where $C$ is the ellipse:

$$x^2+4y^2-2\sqrt{3}x-1=0$$

...on the $xy$-plane and oriented counterclockwise.

A

$\cfrac{\pi}{3}$

B

$\cfrac{4\pi}{3}$

C

$-\cfrac{4\pi}{3}$

D

$-\cfrac{\pi}{3}$

E

$2\pi$