The question is to "prove the reduction formula"
$ \int{ \frac{ x^2 }{ \left(a^2 + x^2\right)^n } dx } = \frac{ 1 }{ 2n-2 } \left( -\frac{x}{ \left( a^2+x^2 \right)^{n-1} } + \int{ \frac{dx}{ \left( a^2 + x^2 \right)^{n-1} } } \right) $
What I got is
Set
$ u = x $
$ du = dx $
$\displaystyle{ dv = \frac{ x }{ \left( a^2 + x^2 \right)^{n} } dx }$
$\displaystyle{ v = \frac{ 1 }{ 2(n+1) \left( a^2 + x^2 \right)^{n+1} } }$
So I got
$ \frac{ 1 }{ 2n+2 } \left( \frac{x}{ \left( a^2 + x^2 \right)^{n+1}} - \int{ \frac{dx}{ \left( a^2+x^2 \right)^{n+1} } } \right) $
Which I believe is correct. They are subtracting from n in the integration step and I'm not sure why