(Line integral) Compute work through vector field

[SiriusAboutAstronomy]

Homework Statement

"Consider the Vector field F(x,y)=<cos(sin(x)+y)cos(x)+e^x, cos(sin(x)+y)+y>. Compute the work done as you traverse the Archimedes spiral (r=θ) from (x,y)=(0,0) to (x,y)=(2∏,0). (Hint: check to see if the vector field is conservative)

Homework Equations

1) F(x,y)=<P,Q> is conservative if $\partial$P/$\partial$y=$\partial$Q/$\partial$x

2) $\int$$_{c}$$\nabla$f$\cdot$dr= f(r(b))-f(r(a))

The Attempt at a Solution

1) The vector field is conservative by equation one: the partial derivative of P and Q with respect to y and x, respectively, are equivalent and equal cos(x)sin(sin(x)+y)

Difficulties:
The field is conservative, which means there exists a function f with ∇f=F. So I could use the fundamental theorem of line integrals, but I don't know how to integrate Q=cos(sin(x)+y)+y with respect to x.
As opposed to this I could try to do a change of variables but I don't know where to start with that.
I think some of my trouble comes from trying to wrap my head around r=θ, does that mean that the position vector r(x,y)=θ? or does it mean that r(r,θ)=<θ,θ>? If the latter is true, do I use polar coordinates? And if I do, how do I put F(x,y) into polar coordinates if r=<θ,θ>

 

[SiriusAboutAstronomy]

or do I need to put F(x,y) into one variable (say theta) and how do I do that? With r(x,y)=<θ,θ>, do I just plug that into F(x,y) or do I need to find f=antiderivative(F(x,y)) and then plug in r(x,y)=<θ,θ> for x and y