# Evaluate line integral

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## 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.

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vela
Staff Emeritus
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.