Question

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

Answer 1

${\int }_3^2\frac{x^3+1}{x(x-1)}dx$

dividing $(x^3+1)$ by $x(x-1)$, we get

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

${\int }_3^2\frac{x^3+1}{x(x-1)}dx={\int }_3^2(x+1-\frac{1}{x})dx$

${[\frac{x^2}{2}+x-log|x|]}_3^2$

$\frac{2^2}{2}+2-log2-\frac{3^2}{2}-3+log3=log\frac{3}{2}-\frac{7}{2}$.