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$F(x,y)=(6x{y}^3+3)i+(9x^2{y}^2+2e^{2y})j$

is conservative by finding a potential function $f$ for $F,$ and use $f$ to compute ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr,$ where $C$ is the curve given by

$r(t)=(2si{n}^{5}t)i+(\frac{2t}{}si{n}^{10}(5t))j$

for $=*=\frac{}{4}$

$f(x,y)=$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr=$