求(1+x^3)分之1的不定积分

Question

求(1+x^3)分之1的不定积分

Answer 1

∫1/(1+x^3)dx
＝1/3×∫[1/(x+1)－(x-2)/(x^2-x+1)]dx
＝1/3×ln|x+1|－1/3×∫(x-2)/(x^2-x+1)dx
＝1/3×ln|x+1|－1/3×∫(x-1/2-3/2)/(x^2-x+1)dx
＝1/3×ln|x+1|－1/6×∫(2x-1)/(x^2-x+1)dx＋1/2×∫1/(x^2-x+1)dx
＝1/3×ln|x+1|－1/6×ln(x^2-x+1)dx＋1/2×∫1/(x-1/2)^2+3/4)dx
＝1/3×ln|x+1|－1/6×ln(x^2-x+1)dx＋1/2×2/√3×arctan[(2x-1)/√3]＋C
＝1/3×ln|x+1|－1/6×ln(x^2-x+1)dx＋1/√3×arctan[(2x-1)/√3]＋C