1. **State the problem:** Find the integral $$\int \frac{x^3 - \sqrt{x}}{x \sqrt{x}} \, dx$$. 2. **Simplify the integrand:** Rewrite the denominator: $$x \sqrt{x} = x \cdot x^{1/2} = x^{3/2}$$. So the integrand becomes: $$\frac{x^3 - x^{1/2}}{x^{3/2}} = \frac{x^3}{x^{3/2}} - \frac{x^{1/2}}{x^{3/2}} = x^{3 - \frac{3}{2}} - x^{\frac{1}{2} - \frac{3}{2}} = x^{\frac{3}{2}} - x^{-1}$$. 3. **Rewrite the integral:** $$\int \left(x^{\frac{3}{2}} - x^{-1}\right) \, dx = \int x^{\frac{3}{2}} \, dx - \int x^{-1} \, dx$$. 4. **Integrate each term separately:** - For $$\int x^{\frac{3}{2}} \, dx$$, use the power rule: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$, where $$n \neq -1$$. Here, $$n = \frac{3}{2}$$, so: $$\int x^{\frac{3}{2}} \, dx = \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} + C = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + C = \frac{2}{5} x^{\frac{5}{2}} + C$$. - For $$\int x^{-1} \, dx$$, recall that: $$\int x^{-1} \, dx = \ln|x| + C$$. 5. **Combine the results:** $$\int \frac{x^3 - \sqrt{x}}{x \sqrt{x}} \, dx = \frac{2}{5} x^{\frac{5}{2}} - \ln|x| + C$$. **Final answer:** $$\boxed{\frac{2}{5} x^{\frac{5}{2}} - \ln|x| + C}$$