Evaluate the following integral over the ellipse ​

The underlying vector field,

F(x, y) = -y/(4x ² + 9y ²) i + x/(4x ² + 9y ²) j,

is conservative, so any integral of F over a closed path is 0.

To establish that F is conservative, we want to find a scalar function f(x, y) whose gradient is equal to F(x, y), which entails solving

Integrating the first equation with respect to x yields

and differentiating with respect to y gives

g(x) is the negative form of f(x)? because whenever its the negative form it reflects its self over the x axis. idk if this is right though.

step-by-step explanation:

your answer would be 12