x^3－x^2＋x-1 =(x-1)(x^2 +1)
let
x/(x^3－x^2＋x-1)≡ A/(x-1)+(Bx+C)/(x^2 +1)
=>
x≡ A(x^2+1)+(Bx+C)(x -1)
x=1, => A =1/2
coef. of x^2
A+B=0
B=-1/2
coef. of constant
A-C =0
C=1/2
x/(x^3－x^2＋x-1)
≡ A/(x-1)+(Bx+C)/(x^2 +1)
≡ (1/2)[1/(x-1)-(x-1)/(x^2 +1)]
∫x/(x^3－x^2＋x-1) dx
=(1/2)∫[ 1/(x-1)-(x-1)/(x^2 +1) ] dx
=(1/2)∫dx/(x-1)-(1/2)∫x/(x^2 +1) dx + (1/2)∫ dx/(1+x^2)
=(1/2)ln|x-1| -(1/4)ln|x^2 +1| +(1/2)arctanx + C
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