As a part of an assignment to calculate the volume of unit n-dimensional ball I got to the following expression, which I believe is true:
$$V_{n}=\left(2\pi\right)^{\frac{n}{2}}\left(n\int_{r=0}^{\infty}\exp\left(\frac{-r^{2}}{2}\right)r^{n-1}\mathrm{d}r\right)^{-1}$$
I need to solve the integral and we were hinted to do it by recurrence. I tried to do it per partes, but I am lost:
Let's denote the n-th integral by $E_n$:
$$E_{n}=\int_{x=0}^{\infty}\exp\left(\frac{-x^{2}}{2}\right)x\cdot x^{n-2}\mathrm{d}x$$
the "base case" is $E_2$: $$E_{2}=\int_{x=0}^{\infty}\exp\left(\frac{-x^{2}}{2}\right)x^{1}\mathrm{d}x=1$$
substitution: $$\begin{pmatrix}\begin{array}{ll} u=x^{n-2} & v'=\exp\left(\frac{-x^{2}}{2}\right)x\\ u'=\left(n-2\right)x^{n-3} & v=-\exp\left(\frac{-x^{2}}{2}\right) \end{array}\end{pmatrix}$$
we get:
$$E_{n}=\int_{x=0}^{\infty}v'\cdot u\mathrm{d}x=\left[uv\right]_{0}^{\infty}-\int_{x=0}^{\infty}v\cdot u'\mathrm{d}x$$
$$ \int_{x=0}^{\infty}v\cdot u'\mathrm{d}x=\int_{x=0}^{\infty}-\exp\left(\frac{-x^{2}}{2}\right)\cdot\left(n-2\right)x^{n-3}\mathrm{d}x=-\left(n-2\right)\int_{x=0}^{\infty}\exp\left(\frac{-x^{2}}{2}\right)\cdot x^{\left(n-1\right)-2}\mathrm{d}x=-\left(n-2\right)E_{n-1}$$
$$ E_{n}=\int_{x=0}^{\infty}v'\cdot u\mathrm{d}x=\left[uv\right]_{0}^{\infty}-\int_{x=0}^{\infty}v\cdot u'\mathrm{d}x=\left[uv\right]_{0}^{\infty}+\left(n-2\right)E_{n-1}$$
$$\left[uv\right]_{0}^{\infty}=\left[\left(x^{n-2}\right)\left(-\exp\left(\frac{-x^{2}}{2}\right)\right)\right]_{0}^{\infty}=0$$
and so:
$$E_{n}=\left(n-2\right)E_{n-1}=\left[\left(n-2\right)\left(n-3\right)\cdot\cdots\cdot3\right]E_{2}=\frac{\left(n-2\right)!}{2}\cdot1=\frac{\left(n-2\right)!}{2}$$
and finally:
$$V_{n}=\left(2\pi\right)^{\frac{n}{2}}\cdot\left(n\frac{\left(n-2\right)!}{2}\right)^{-1}=\left(2\pi\right)^{\frac{n}{2}}\cdot\frac{2}{n\left(n-2\right)!}=\pi^{\frac{n}{2}}\frac{2^{\frac{n}{2}+1}}{n\left(n-2\right)!}$$
which does not seem to be the correct formula for $V_n$, because i.e. for $n=3$ we get $\frac{4}{3} \sqrt{2} \pi^\frac{3}{2}$.
If only you could spot the mistake or mistakes, I would be very grateful.
The big mistake you are making here is:
$E_n = (n-2)E_{n-1}$
The index on the right hand side is off. Remember:
$E_n = \int \exp(-x^2/2) x^{n-1} dx$,
so
$E_{n-1} = \int \exp(-x^2/2) x^{n-2} dx.$
(Hint: You should not be able to end up with $E_2$ all the time.)