Question

$\int \frac{3x+1}{(x^3-x^2-x+1)}dx$

Answer 1

$I=\int \frac{3x+1}{x^3-x^2-x+1}dx$

Using partial fraction

$\frac{3x+1}{x^3-x^2-x+1}=\frac{1}{2(x-1)}+\frac{2}{(x-1^2)}-\frac{1}{2(x+1)}$

$I=\int [\frac{1}{2(x-1)}+\frac{2}{(x-1{)}^2}-\frac{1}{2(x+1)}]dx$

$I=\frac{1}{2}\log (x-1)-\frac{2}{(x-1{)}^2}-\frac{1}{2}\log (x+1)$