If $X, \ Y,\ Z$ are independent standard normal random variables, I'm trying to find the distribution of $X^2 +Y^2 + Z^2$ using spherical coordinates.
Taking $$x=rsin\theta \cos\phi, \ \ y = rsin\theta \sin\phi, \ \ z = cos\theta$$
$$r \in [0,\sqrt t] \ \ \theta \in [0, \pi] \ \ \phi \in [0,2\pi] $$
I'm going to compute the cdf to be able to tell afterwards which distribution it follows.
Therefore after using that the joint pdf becomes the product of marginals because of independence we get
$$ P(X^2 +Y^2 + Z^2 \leq t) = \int_0^{2\pi}\int_0^\pi\int_0^{\sqrt t}\frac{1}{(2\pi)^{3/2}}e^{-r^2/2}r^2sin\theta drd\theta d\phi = \int_0^{\sqrt t}\frac{2}{(2\pi)^{1/2}}e^{-r^2/2}r^2 dr$$ but here, I don't know how to compute this integral.
If $X_1, ..., X_n$ are i.i.d. standard Gaussians then the distribution of $X_1^2 + ... + X_n^2$ is known as the $\chi_n^2$-distribution. Calculating the cdf of the $\chi^2$ distributions is quite involved and includes special functions. See the wiki page for more information.