Integration over D-dimensional euclidean space

Question

I have to show that in D-dimensional Euclidean space:

\begin{align} \int d^D q=\frac{2\pi^{D/2}}{\Gamma(\frac{D}{2})}\int d q^{D-1} \end{align}

By the way shouldn't it be $\int dq q^{D-1}$ if it should make any sense?

I was given the hint to look at the gaussian integral:

\begin{align} \int d^D q e^{-q^2} \end{align}

But I don't know if I should write it out in D integrals, that is assume

\begin{align} q^2=(x_1^2+x_2^2+...+x_D^2) \end{align}

I don't know if this approach is useful. I hope someone can help.