Answer
$\frac{1}{2}\cdot[{e^1}-1]\approx 0.8591 $,
Work Step by Step
Let $u=x^2$, therefore $\frac{du}{dx}=2x$, therefore$\int_{0}^{1} (x\cdot e^{x^2}) dx=\frac{1}{2}\cdot{ \int_{0}^{1} (e^u}) du=\frac{1}{2}\cdot[{e^u}]_{0}^{1}=\frac{1}{2}\cdot[{e^1}-e^0]=\frac{1}{2}\cdot[{e^1}-1]\approx 0.8591 $,