Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $
Express recurrence relation of the integral
$$ I_n=\int\frac{dx}{(1+x^2)^n} $$
[My Answer]
$$ I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx $$
$$ I_n=I_{n-1}-\int x\cdot\frac{x}{(1+x^2)^n}dx $$
$$ I_n=I_{n-1}-\frac{x}{2(1-n)(x^2+1)^{n-1}}+\frac{1}{2(1-n)}I_{n-1} $$
$$ I_n=\frac{2n-3}{2(n-1)}I_{n-1}+\frac{x}{2(n-1)(x^2+1)^{n-1}} \ \ \ \ (n>1) $$
$$ I_1=\arctan(x) $$
Is my answer correct?
Yes, the answer is correct (up to a constant, but it does not not change the idea).