Show that \(\int_{0}^{1} x \mathrm{e}^{-x^{2}} \mathrm{~d} x=\frac{1}{2}-\frac{1}{2 \mathrm{e}}\).

Let \(I_{n}=\int_{0}^{1} x^{n} \mathrm{e}^{-x^{2}} \mathrm{~d} x\). Show that \(I_{2 n+1}=n I_{2 n-1}-\frac{1}{2 \mathrm{e}}\) for \(n \geqslant 1\).

Find the exact value of \(I_{7}\).