Need help with some aspects of this problem (particularly the last part). Would greatly appreciate any help offered.

'For $n \geq 0$, let

$I_n=\int\limits_0^1{x}^{n}(1-x)^n\, \mathrm{d}x $

For $n \geq 1$, show, by means of a substitution, that

$\int\limits_0^1{x}^{n-1}(1-x)^n\, \mathrm{d}x=\int\limits_0^1{x}^n(1-x)^{n-1}\, \mathrm{d}x $

and deduce that

$2\int\limits_0^1{x}^{n-1}(1-x)^n\, \mathrm{d}x=I_{n-1}$

Show also, for $n \geq 1$, that

$I_n=\frac{n}{n+1}\int\limits_0^1{x}^{n-1}(1-x)^{n+1}\, \mathrm{d}x $

and hence that

$I_n=\frac{n}{2(2n+1)}I_{n-1}$.'