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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?

  • 0
    The answer you got is correct.2012-10-31

1 Answers 1

1

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