I would need a hint to solve the following problem.
Let $d \varphi$ be denote the normalised uniform measure on $\mathbb{S}^{N-1}$, which is the $N-1$ dimensional sphere, where $N \in \mathbb{N}_{>0}$. How to prove that, for arbitrary non-negative integers, $n_1$, $\ldots$, $n_N$, we have that $$ \int_{\mathbb{S}^{N-1}} d \varphi \, \, {\big ( \varphi^1 \big )}^{2 n_1} \ldots {\big (\varphi^N \big )}^{2 n_N} = \frac{ \Gamma(N/2)}{2^n \Gamma(n + N/2)} \prod_{i=1}^{N} (2n_i - 1)!!, $$ where $n= n_1 + \ldots + n_N$ and we used the notation $\varphi =( \varphi^1, \ldots, \varphi^N) \in \mathbb{R}^N$.
Here $\Gamma(x)$ is the *Gamma function" and "!!" is the double factorial. These objects are defined here: https://en.wikipedia.org/wiki/Double_factorial; https://en.wikipedia.org/wiki/Gamma_function.
Clearly $$ I\cdot\omega_{N-1}\cdot\int_0^\infty r^{2n}\exp\left(-\frac12r^2\right)\,r^{N-1}\,\mathrm{d}r=\prod_{i=1}^N\int_{-\infty}^\infty x_i^{2n_i}\exp\left(-\frac12x_i^2\right)\,\mathrm{d}x_i $$ where $\omega_{N-1}=\dfrac{2\pi^{N/2}}{\Gamma(N/2)}$ is the volume of the standard $\mathbb{S}^{N-1}$ and $$ I:=-\!\!\!\!\!\!\!\int_{\mathbb{S}^{N-1}} \left(\varphi^1\right)^{2 n_1} \dots \left(\varphi^N\right)^{2 n_N}\,\mathrm{d}\mathcal{H}^{N-1}. $$ Now you can compute these integrals.