Finding parametrizattion for a Line Integral

I've been asked to evaluate the line integral

$\displaystyle \int (10x^4 - 2xy^3)dx - 3x^2y^2dy $

Over the curve

$\displaystyle x^4 - 6xy^3 - 4y^2 = 0 $

Between the points (0,0) and (2,1).

What I'm trying to do is finding a parametrization x(t) from the given curve and then put that into the integral so I can solve it.

However I can't seem to make it work. I tried solving x for y, so I can say x=t and y is a function of t. Or the other way round would work as well. But the crossterm $\displaystyle 6xy^3$ gets in the way. And the powers of both x and y aren't right to use the ABC formula.

Could anyone help me out on what to do here?
Thanks in advance

Edit: Apologies for the spelling mistake in the thread title. Can't edit it, it seems.
Edit 2: Got one sign in the integral wrong.

Can I ask to check on the sign of one of your terms in the integral

i.e. is it $\displaystyle
\int \limits_{c} (10x^4 {\color{red}{+}}\, 2xy^3)dx + 3x^2y^2dy$

Ah, I did write it down incorrectly, but it's not that either. Correct version is:

$\displaystyle
\int (10x^4 - 2xy^3)dx - 3x^2y^2dy
$

Quote:

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Originally Posted by AbAeterno

Ah, I did write it down incorrectly, but it's not that either. Correct version is:

$\displaystyle
\int (10x^4 - 2xy^3)dx - 3x^2y^2dy
$

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Note that $\displaystyle 2xy^3dx + 3 x^2 y^2 dy = d(x^2y^3)$.

Quote:

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Originally Posted by Danny

Note that $\displaystyle 2xy^3dx + 3 x^2 y^2 dy = d(x^2y^3)$.

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Okay. So I can write that down in the integral and get

$\displaystyle
\int 10x^4 dx - d(x^2y^3)
$
But then what? Still doesnt help me find a parametrisation as far I can see. Or is that not what I'm supposed to do?

Edit: oh wait.. so now I don't need to find a x(t) and y(t) anymore, but an x(t) and x^2y^3(t)?

Quote:

---

Originally Posted by AbAeterno

Okay. So I can write that down in the integral and get

$\displaystyle
\int 10x^4 dx - d(x^2y^3)
$
But then what? Still doesnt help me find a parametrisation as far I can see. Or is that not what I'm supposed to do?

---

Since the vector field is conservative, the parameterization doesn't matter. So

$\displaystyle
\int \limits_c 10x^4 dx - d(x^2y^3) =
\int \limits_c d(2x^5 - x^2y^3) = \left. 2x^5 - x^2y^3 \right|_{(0,0)}^{(2,1)}
$

Ooh.. I totally didn't see that. Thanks!