Show $$\mathbf{F}(x,y) = (6 x y^3 + 2)\mathbf{i} + (9 x^2 y^2 + 2e^{2y})\mathbf{j}$$ is conservative by finding a potential function $f$ for $\mathbf{F}$, and use $f$ to compute $\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}$, where $C$ is the curve given by $$\mathbf{r}(t) = ( 2\sin^{3} t ) \mathbf{i} + \left( \frac{2t}{\pi}\sin^{10} (5t) \right) \mathbf{j}$$ for $0 \le t \le \pi/2$.

$f(x,y) =$

$\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r} =$