# Deriving Reduction formula - Indefinite integration using integration by parts

I was working on finding the reduction formula for :

$\int \frac{dx}{(x^2+a^2)^n}$ By using integration by parts formula ( $\int f(x) g(x) dx = f(x) \int g(x)dx -\int (f'(x)\int g(x))$ considering 1.dx as second function :

Let $I_n$ = $\int \frac{dx}{(x^2+a^2)^n} = \frac{dx}{(x^2+a^2)^n}.\int dx -\int \{\frac{dx}{(x^2+a^2)^n}.\int \}dx$

=$\frac{dx}{(x^2+a^2)^n}-\int (x^2+a^2)^{n-1}.2x^2dx$

=$\frac{dx}{(x^2+a^2)^n} +2n\int \frac{x^2}{(x^2+a^2)^{n+1}}dx$

=$\frac{dx}{(x^2+a^2)^n} + \frac{2n}{(x^2+a^2)}\int \frac{dx}{(x^2+a^2)^{n-1}}-\frac{dx}{(x^2+a^2)^n}$

Therefore , $I = \frac{x}{(x^2+a^2)^n}+2nI_n-2n^2I_{n+1}$

Please guide further how to proceed and conclude..Thanks

If $$I_n=\int \frac{dx}{(x^2+a^2)^n}$$ then setting $$u=\frac{1}{(x^2+a^2)^n},~~dv=dx$$ leads us to have $$du=\frac{-2nxdx}{(x^2+a^2)^{n+1}},~~v=x$$ and the method of integrating by parts gives us $$I_n=\frac{x}{(x^2+a^2)^n}+2n\int\frac{x^2dx}{(x^2+a^2)^{n+1}}=...=\frac{x}{(x^2+a^2)^n}+2nI_n-2na^2I_{n+1}$$ Now we see the integrating of $I_{n+1}$ will be dependent to integrating of $I_n$ and....