展开全部
分式裂项：
3/(x³+1) = 3/[(x+1)(x²-x+1)
= 1/(x+1) - (x-2)/(x²-x+1)
= 1/(x+1) - (x-1/2)/(x²-x+1) + (3/2)/[(x-1/2)²+(√3/2)²]
∴ ∫3/(x³+1) dx = ∫{1/(x+1) - (x-1/2)/(x²-x+1) + (3/2)/[(x-1/2)²+(√3/2)²] } dx
= ∫1/(x+1) dx - ∫(1/2)/(x²-x+1) d(x²-x+1) + ∫ (3/2)/[(x-1/2)²+(√3/2)²] d(x-1/2)
= ln|x+1| - (1/2)ln(x²-x+1) + 3/2*2/√3*arctan(2x/√3)
= ln|x+1| - (1/2)ln(x²-x+1) + (√3)arctan[(2√3)x/]
3/(x³+1) = 3/[(x+1)(x²-x+1)
= 1/(x+1) - (x-2)/(x²-x+1)
= 1/(x+1) - (x-1/2)/(x²-x+1) + (3/2)/[(x-1/2)²+(√3/2)²]
∴ ∫3/(x³+1) dx = ∫{1/(x+1) - (x-1/2)/(x²-x+1) + (3/2)/[(x-1/2)²+(√3/2)²] } dx
= ∫1/(x+1) dx - ∫(1/2)/(x²-x+1) d(x²-x+1) + ∫ (3/2)/[(x-1/2)²+(√3/2)²] d(x-1/2)
= ln|x+1| - (1/2)ln(x²-x+1) + 3/2*2/√3*arctan(2x/√3)
= ln|x+1| - (1/2)ln(x²-x+1) + (√3)arctan[(2√3)x/]
已赞过
已踩过<
评论
收起
你对这个回答的评价是？
推荐律师服务：
若未解决您的问题，请您详细描述您的问题，通过百度律临进行免费专业咨询
