1. ## Reduction Method Question

2. First off, there's a typo (an n is missing)

$\int(x^2+a^2)^ndx$ if $u = (x^2+a^2)^n,\;\; dv= dx$ so $du = 2 n x (x^2+a^2)^{n-1},\;\;v = x$ so

$I_n = \int(x^2+a^2)^ndx$
$= x(x^2+a^2)^n - 2n \int x^2 (x^2+a^2)^{n-1}dx$
$= x(x^2+a^2)^n - 2n \int (x^2+a^2-a^2) (x^2+a^2)^{n-1}dx$
$= x(x^2+a^2)^n - 2n \int (x^2+a^2) (x^2+a^2)^{n-1}dx + 2na^2 \int (x^2+a^2)^{n-1}dx$
$I_n = x(x^2+a^2)^n - 2n \underbrace{\int (x^2+a^2)^{n}dx}_{I_n} + 2na^2 \underbrace{\int (x^2+a^2)^{n-1}dx}_{I_{n-1}}$
so

$(1+2n) I_n = x(x^2+a^2)^n +2na^2 I_{n-1}$

You circled a term and called it $I_{n-1}$ (I change it for this conversation). This suggests that there is an $I_{n}$. Second, put $a=0,\;\;n=2$ - your formula doesn't work!