[tex]\int (\frac{4x^3-x}{x-2}) dx =\int \frac{4x^3}{x+2}dx-\int \frac{x}{x+2}dx
\\ =\ \textgreater \ \int \frac{4x^3}{x+2}dx =4\cdot \int \:\frac{x^3}{x+2}dx =4\cdot \int \:\frac{\left(u-2\right)^3}{u}du \:\
\\ \boxed{u=x+2}
\\=4\cdot \int \:u^2-6u+12-\frac{8}{u}du
\\ =4\left(\int \:u^2du-\int \:6udu+\int \:12du-\int \frac{8}{u}du\right)
\\ =4\left(\frac{\left(x+2\right)^3}{3}-3\left(x+2\right)^2+12\left(x+2\right)-8\ln \left|x+2\right|\right) [/tex]
[tex]=-12x^2-x+\frac{4}{3}\left(x+2\right)^3-30\ln \left|x+2\right|+46+C[/tex]
