Question: Let I n = x n dx/x 2 + 1. (a) Prove that In = x n1 1 I n2 . (b) Use

Let In = xn dx/x2 + 1.

(a) Prove that In = xn−1  − 1 − In−2.
(b) Use (a) to calculate In for 0 ≤ n ≤ 5.
(c) Show that, in general,

12n+1 = 12n = xn 2n x2n-2 2n - 2 12 + (1)- + (1)1 In(x + 1) + C 2 x2n-1 x2n-3 2n - 1 2n-3 + (-1)"x + (-1)"

12n+1 = 12n = xn 2n x2n-2 2n - 2 12 + (1)- + (1)1 In(x + 1) + C 2 x2n-1 x2n-3 2n - 1 2n-3 + (-1)"x + (-1)" tan x + C

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