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Integrate the rational functions
$\frac{3 x+5}{x^3-x^2-x+1}$
$\frac{3 x+5}{x^3-x^2-x+1}$
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Let $\frac{3 x+5}{x^2(x-1)-1(x-1)}$
$=\frac{3 x+5}{(x-1)^2(x+1)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x+1}$
$\Rightarrow 3 x+5=\mathrm{A}(x-1)(x+1)+\mathrm{B}(x+1)+\mathrm{C}(x-1)$
Put $x=1,-1,0 \Rightarrow \mathrm{B}=4, \mathrm{C}=\frac{1}{2},=\mathrm{A}=-\frac{1}{2}$
$\therefore \mathrm{I}=-\frac{1}{2} \int \frac{d x}{(x-1)}+4 \frac{d x}{(x-1)^2}+\frac{1}{2} \int \frac{d x}{x+1}$
$=\frac{1}{2} \log \left|\frac{x+1}{x-1}\right|-\frac{4}{x-1}+\mathrm{C}$
$=\frac{3 x+5}{(x-1)^2(x+1)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x+1}$
$\Rightarrow 3 x+5=\mathrm{A}(x-1)(x+1)+\mathrm{B}(x+1)+\mathrm{C}(x-1)$
Put $x=1,-1,0 \Rightarrow \mathrm{B}=4, \mathrm{C}=\frac{1}{2},=\mathrm{A}=-\frac{1}{2}$
$\therefore \mathrm{I}=-\frac{1}{2} \int \frac{d x}{(x-1)}+4 \frac{d x}{(x-1)^2}+\frac{1}{2} \int \frac{d x}{x+1}$
$=\frac{1}{2} \log \left|\frac{x+1}{x-1}\right|-\frac{4}{x-1}+\mathrm{C}$
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