$($D$1)$ Let $F(x,y)=($:$3+2xy,x^2-3y^2$:$).$

a$.[3$ pts$]$ Determine whether $F$ is conservative. Show all work that leads to your conclusion.

b$.[5$ pts$]$ If $F$ is conservative, find a potential function an use it along with the Fundamental Theorem for Line Integrals to evaluate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr,$ where $C$ is the curve given by $r(t)=($:$e^{t}sint,e^{t}cost$:$),0t.$ If $F$ is not conservative, evaluate the line integral directly.