Path Independence in Line Integrals: Simplifying Evaluation | Problem Attached

[DryRun]

Homework Statement

I have attached the problem to the post.

Homework Equations

Properties of line integral. Path independence.

The Attempt at a Solution

I have shown that the path is independent, as:
\partial P/\partial y = \partial Q/\partial x
The problem is with the parametrization. I found $dx/dt$ and $dy/dt$ and replaced into the line integral as well as x and y, so i have the line integral in terms of $t$ only. But the expansion becomes such a mess. I don't know if there's some simplification to be done, before integrating w.r.t.t. If not, then I'm stuck. I have a doubt that the path being independent has something to do with the simplification of the evaluation of line integral, but i can't figure how.

 

[vela]

It looks like you need to either evaluate the integral using the original contour numerically or choose a different path to make the integral doable.

 

[gopher_p]

... or you could use the Fundamental Theorem for Line Integrals.

 

[DryRun]

I got the answer key for this today and it involves using the 3rd theorem of line integrals, which converts the line integral into a function, $\phi (x,y)$ and then just evaluate that function over the limits by calculating the two sets of points in terms of x and y. No integration required! At least, not to get the final solution. It's surprising, as the answer is very short.