Integration by parts. Need to show that:
If $I_n(x)=\int_{0} ^{x}t^n(t^2+a^2)^{-\frac{1}{2}}dt$
Then: $nI_n(x) = x^{n-1}\sqrt{x^2+a^2}-(n-1)a^2I_{n-2(x)}$ if $x\geq2$
I can get to the point where:
$I_n=\frac{x^{n+1}}{n+1}(x^2+a^2)^{-\frac{1}{2}}+\frac{1}{n+1}\int_0 ^x (t^nt^2+a^2t^n-a^2t^n)(t^2+a^2)^{-\frac{3}{2}}dt$
$=~\frac{x^{n+1}}{n+1}(x^2+a^2)^{-\frac{1}{2}}+ \frac{1}{n+1}I_n-\frac{1}{n+1}\int_0 ^x a^2t^n(t^2+a^2)^{-\frac{3}{2}}dt$
$=>~nI_{n}=x^{n+1}(x^2+a^2)^{-\frac{1}{2}}-a^2I_{n-2}+a^4\int t^{n-2}(t^2+a^2)^{-\frac{3}{2}}dt$
Now, is this a good start or should I have taken another route? Because I can not find a way out :/