Answer
$\frac{11}{8}-\frac{1}{e}$
Work Step by Step
We know that:
$F(x,y)=(e^{x-1},xy)$ and $ C$ is $r(t)=(t^2,t^3), $$ 0 \leq t \leq1$
$$\int_{c} F.dr= \int_{c} F(r(t)).r'(t) dt $$ $r'(t)=2t i +3t^2 j$
$ \displaystyle\int _{0}^{1} (e^{t^{2}-1}i + t^5 j). (2t i + 3t^2 j)$
$= \displaystyle \int _{0}^{1} (2te^{t^{2}-1} + 3t^7 ) dt$
$= \left(e^{t^{2}-1} + \frac{3}{8}t^8\right) \bigg |_{0}^{1} $
$= \frac{11}{8}-\frac{1}{e}$