Solution
= Solution
Start with the supplied <Gaussian integral> $I(0)=\sqrt\pi/2$ and part (i)'s $I(1)=1/2$. Repeated use of the <Gaussian moment recurrence> gives
$$
\boxed{I(2)=\frac{\sqrt\pi}{4},\quad I(3)=\frac12,\quad I(4)=\frac{3\sqrt\pi}{8},\quad I(5)=1,\quad I(6)=\frac{15\sqrt\pi}{16}.}
$$
The even indices inherit the $\sqrt\pi$ factor from $I(0)$, while the odd indices inherit rational values from $I(1)$.