求解(x^2-x+1)分之(x+1)的积分过程

Question

求解(x^2-x+1)分之(x+1)的积分过程

Answer 1

∫ (x + 1) / (x^2 - x + 1) d x
= ∫ (x - 1) / (x^2 - x + 1) d x + ∫ 2 / (x^2 - x + 1) d x
= ∫ 1 / (x^2 - x + 1) d (x^2 - x + 1)
+ 2 ∫ 1 / [ (x - 1 / 2)^2 + (√3 / 2)^2 ] d (x - 1 / 2)
= ∫ 1 / t d t + 2 ∫ 1 / [ u^2 + (√3 / 2)^2 ] d u
= ln t + C1 + 2 * 2 /√3 * arctan (2 u / √3) + C2
= ln (x^2 - x + 1) + C1 + 4 arctan [2 (x - 1 / 2) / √3] /√3 + C2
= ln (x^2 - x + 1) + 4 arctan [2 √3 (x - 1 / 2) / 3] /√3 + C