(5) [Marked] Consider the downward-oriented (in the negative y-direction) quintic curve, \[ C=\left\{(x, y) \in \mathbb{R} \times(-1,1) \mid x=y^{5}+y+2\right\}, \] and consider the vector field $\mathbf{F}$ on $\mathbb{R}^{2}$ given by \[ \mathbf{F}(x, y)=(x-y, x+y)_{(x, y)} . \] (a) Give an injective parametrisation $\gamma$ of $\mathrm{C}$ such that the image of $\gamma$ differs from $\mathrm{C}$ by only a finite number of points. Which orientation of $\mathrm{C}$ does $\gamma$ generate? (b) Compute the (curve) integral of $\mathbf{F}$ over the curve $\mathbf{C}$.