1/(x^3-1)求积分

Question

1/(x^3-1) 求积分

Answer 1

1/(x^3-1)=1/(x-1)(x^2+x+1)=1/3*1/(x-1)-1/3 [(x+2)/(x^2+x+1)]
所以
原式=1/3∫1/(x-1)dx-1/3∫(x+2)/(x²＋x+1)dx
=1/3ln|x-1|-1/3∫ (x+1/2+3/2)/(x²＋x+1)dx
=1/3ln|x-1|-1/3∫ (x+1/2)/(x²＋x+1)dx-(1/2)∫1/(x²＋x+1)dx
=1/3ln|x-1|-1/6 ln(x²＋x+1)-1/2 ∫1/[(x+1/2)²+3/4]dx
=1/3ln|x-1|-1/6 ln(x²＋x+1)-1/2×1/(√3/2)arctan[(x+1/2)/(√3/2) ]+c
下面自己化简吧