求有理函数的不定积分§X/(X^3－1)dx
问题描述：
求有理函数的不定积分§X/(X^3－1)dx
答
∫x/(x³-1) dx
=-1/3*∫(x-1)/(x²+x+1)+1/3*∫1/(x-1)
=-1/6*∫(2x+1)/(x²+x+1)+1/2*∫1/(x²+x+1)+1/3*ln(x-1)
=-1/6*ln(x²+x+1)+1/√3*arctan[(2X+1)/√3]+1/3*ln(x-1)
答
用换元法，令x=1/t带入之后化简得￡1／(t*2-1)dt=￡0.5(1/(t-1))dt-￡0,5(1/(t+1))dt=0.5ln(t-1)/(t+1)=0.5ln(1-x)/(1+x)
答
Sx/(x^3-1)dx=1/3S*(1/(x-1)-(x-1)/(x^2+x+1))dx
=1/3*ln|x-1|-1/6*ln(x^2+x+1)+2根3/3*arctan(x/根3+1/根3)+c