nd the most general antiderivative of the function. (Check your answ \[ f(x)=\frac{2 x^{4}+3 x^{3}-x}{x^{3}}, \quad x>0 \]

To find the most general antiderivative of the function \( f(x) = \frac{2x^4 + 3x^3 - x}{x^3} \), we first simplify the function: \[ f(x) = \frac{2x^4}{x^3} + \frac{3x^3}{x^3} - \frac{x}{x^3} = 2x + 3 - \frac{1}{x^2}. \] Now, we can find the antiderivative by integrating each term: \[ \int f(x) \, dx = \int (2x + 3 - \frac{1}{x^2}) \, dx. \] Calculating the integrals gives us: \[ \int 2x \, dx = x^2, \] \[ \int 3 \, dx = 3x, \] \[ \int -\frac{1}{x^2} \, dx = \frac{1}{x}. \] Putting it all together, we have: \[ \int f(x) \, dx = x^2 + 3x + \frac{1}{x} + C, \] where \( C \) is the constant of integration. Thus, the most general antiderivative of the function is: \[ F(x) = x^2 + 3x + \frac{1}{x} + C. \] Now you can check your integration by differentiating \( F(x) \) to see if it returns to the original function \( f(x) \)!