Use a change of variables or the table to evaluate the following definite integral. \( \int_{2}^{3} x^{2} e^{x^{3}}-8 \) \( d x \) Click to view the table of general integration formulas. \( \frac{3}{3} x^{2} e^{x^{3}-8} d x=\square \) (Type an exact answer.)

To solve the integral \( \int_{2}^{3} (x^{2} e^{x^{3}} - 8) \, dx \), first split it into two parts: \[ \int_{2}^{3} x^{2} e^{x^{3}} \, dx - \int_{2}^{3} 8 \, dx \] For the first part, we can use the change of variable \( u = x^{3} \), which gives \( du = 3x^{2} \, dx \) or \( \frac{du}{3} = x^{2} \, dx \). We also need to adjust the limits: when \( x = 2 \), \( u = 8 \); when \( x = 3 \), \( u = 27 \). Thus, the integral becomes: \[ \int_{8}^{27} e^{u} \cdot \frac{1}{3} \, du \] \[ = \frac{1}{3} \left( e^{27} - e^{8} \right) \] Next, calculate the second part: \[ \int_{2}^{3} 8 \, dx = 8 \cdot (3-2) = 8 \] Now combine the two results: \[ \frac{1}{3} (e^{27} - e^{8}) - 8 \] Therefore, the final answer for the definite integral is: \[ \frac{1}{3}(e^{27} - e^{8}) - 8 \]