Evaluate the definite integral $\int_{0}^{1} x e^{x^{2}} d x$.

(N/A) Let $I = \int_{0}^{1} x e^{x^{2}} d x$.
Substitute $x^{2} = t$,which implies $2x \, dx = dt$ or $x \, dx = \frac{1}{2} dt$.
Change the limits of integration:
When $x = 0$,$t = 0^{2} = 0$.
When $x = 1$,$t = 1^{2} = 1$.
Substituting these into the integral,we get:
$I = \int_{0}^{1} e^{t} \cdot \frac{1}{2} dt = \frac{1}{2} \int_{0}^{1} e^{t} dt$.
The integral of $e^{t}$ is $e^{t}$.
Applying the limits:
$I = \frac{1}{2} [e^{t}]_{0}^{1} = \frac{1}{2} (e^{1} - e^{0})$.
Since $e^{0} = 1$,we have:
$I = \frac{1}{2} (e - 1)$.