∫dx/x(x^3+1)

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∫dx/x(x^3+1)∫dx/x(x^3+1)∫dx/x(x^3+1)∫dx/[x(x^3+1)]=∫dx/[x(x+1)(x^2-x+1)]let1/[x(x^3+1)]≡A/x+B/(x+1)+

∫dx/x(x^3+1)
∫dx/x(x^3+1)

∫dx/x(x^3+1)
∫dx/[x(x^3+1)]
=∫dx/[x(x+1)(x^2-x+1)]
let
1/[x(x^3+1)]≡A/x +B/(x+1) + (Cx+D)/(x^2-x+1)
=>
1≡A(x+1)(x^2-x+1) +Bx(x^2-x+1)+ (Cx+D)x(x+1)
x=0, A=1
x=-1, B=-1/3
coef. of x^3
A+B+C=0
C=-2/3
coef. of x
B+D =0
D= 1/3
ie
1/[x(x^3+1)]≡1/x -(1/3)[1/(x+1)] -(1/3) (2x-1)/(x^2-x+1)
∫dx/[x(x^3+1)]
=∫{1/x -(1/3)[1/(x+1)] -(1/3) (2x-1)/(x^2-x+1)} dx
= lnx - (1/3)ln|x+1| - (1/3)ln|x^2-x+1| + C

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