$\int (\frac{2}{x^3}+\frac{4}{x^2}-\frac{3}{x^3}+\frac{2}{x})dx=$

$\int (\frac{2}{x^3}+\frac{4}{x^2}-\frac{3}{x^3}+\frac{2}{x})dx$

$=\int \frac{2}{x^3}+\int \frac{4}{x^2}-\int \frac{3}{x^3}+\int \frac{2}{x}$

$=2\int x^{-4}+4\int x^{-2}-3\int x^{-3}+2\int \frac{1}{x}$

$=2\left(\frac{x^{-4+1}}{-4+1}\right)+4\left(\frac{x^{-2}+1}{-2+1}\right)-3\left(\frac{x^{-3+1}}{3+1}\right)+2\log x+c$

$=\frac{2x^{-3}}{-3}+\frac{4\left(x^{-1}\right)}{-1}-\frac{3\left(x^{-2}\right)}{-2}+2\log x+c$

$=\frac{-2}{3x^3}-\frac{4}{x}+\frac{3}{2x^2}+2\log x+c$