1/[x(1 x)(1 x x^2)]≡A/x B/(x 1) (Cx D)/(x^2 x 1)
=>
1≡A(1 x)(1 x x^2) Bx(1 x x^2) (Cx D)x(1 x)
x=0,=>A=1/3
x=-1,=>B=-1
A B C=0
1/3-1 C=0
C=-2/3
x=1
6A 3B 2(C D)=1
2-3-4/3 2D=1
D=5/3
1/[x(1 x)(1 x x^2)]
≡(1/3)(1/x)-1/(x 1) (1/3)[(-2x 5)/(x^2 x 1)]
∫dx/[x(1 x)(1 x x^2)]
=∫(1/3)(1/x)-1/(x 1) (1/3)[(-2x 5)/(x^2 x 1)dx
=(1/3)ln|x|-ln|x 1| (1/3)∫(-2x 5)/(x^2 x 1)dx
=(1/3)ln|x|-ln|x 1|-(1/3)∫(2x 1)/(x^2 x 1)dx 2∫dx/(x^2 x 1)dx
=(1/3)ln|x|-ln|x 1|-(1/3)ln|x^2 x 1| 2∫dx/(x^2 x 1)dx
=(1/3)ln|x|-ln|x 1|-(1/3)ln|x^2 x 1| (4√3/3)arctan[(2x 1)/√3] C
x^2 x 1=(x 1/2)^2 3/4
x 1/2=(√3/2)tanu
dx=(√3/2)(secu)^2du
∫dx/(x^2 x 1)
=∫(√3/2)(secu)^2du/[(3/4)(secu)^2]
=(2√3/3)u C
=(2√3/3)arctan[(2x 1)/√3] C