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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?

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The answer you got is correct. –  André Nicolas Oct 31 '12 at 12:06

1 Answer 1

Yes, the answer is correct (up to a constant, but it does not not change the idea).

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