\( \int ( 3 x ^ { \frac { 1 } { 3 } } + 5 x ^ { - \frac { 1 } { 3 } } + 9 ) d x = \square \)

To evaluate the indefinite integral: \[ \int \left( 3x^{\frac{1}{3}} + 5x^{-\frac{1}{3}} + 9 \right) \, dx \] We'll integrate each term separately. 1. **Integrate \( 3x^{\frac{1}{3}} \):** \[ \int 3x^{\frac{1}{3}} \, dx = 3 \cdot \frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} = 3 \cdot \frac{x^{\frac{4}{3}}}{\frac{4}{3}} = \frac{9}{4}x^{\frac{4}{3}} \] 2. **Integrate \( 5x^{-\frac{1}{3}} \):** \[ \int 5x^{-\frac{1}{3}} \, dx = 5 \cdot \frac{x^{-\frac{1}{3} + 1}}{-\frac{1}{3} + 1} = 5 \cdot \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = \frac{15}{2}x^{\frac{2}{3}} \] 3. **Integrate \( 9 \):** \[ \int 9 \, dx = 9x \] 4. **Combine all the results and add the constant of integration \( C \):** \[ \int \left( 3x^{\frac{1}{3}} + 5x^{-\frac{1}{3}} + 9 \right) \, dx = \frac{9}{4}x^{\frac{4}{3}} + \frac{15}{2}x^{\frac{2}{3}} + 9x + C \] **Final Answer:** \[ \frac{9}{4}\, x^{4/3} + \frac{15}{2}\, x^{2/3} + 9\, x + C \]