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 evaluate the definite integral \[ \int_{2}^{3} \left( x^{2} e^{x^{3}} - 8 \right) \, dx, \] we can split it into two separate integrals: \[ \int_{2}^{3} x^{2} e^{x^{3}} \, dx - \int_{2}^{3} 8 \, dx. \] ### 1. Evaluating \(\int_{2}^{3} x^{2} e^{x^{3}} \, dx\): Let's use a substitution method: - Let \( u = x^{3} \). - Then, \( \frac{du}{dx} = 3x^{2} \) or \( \frac{1}{3} du = x^{2} dx \). Substituting into the integral: \[ \int x^{2} e^{x^{3}} \, dx = \frac{1}{3} \int e^{u} \, du = \frac{1}{3} e^{u} + C = \frac{1}{3} e^{x^{3}} + C. \] Evaluating from \( 2 \) to \( 3 \): \[ \left. \frac{1}{3} e^{x^{3}} \right|_{2}^{3} = \frac{1}{3} \left( e^{27} - e^{8} \right). \] ### 2. Evaluating \(\int_{2}^{3} 8 \, dx\): This is a straightforward integral: \[ \int_{2}^{3} 8 \, dx = 8x \Big|_{2}^{3} = 8(3) - 8(2) = 24 - 16 = 8. \] ### Combining Both Results: Putting it all together: \[ \int_{2}^{3} \left( x^{2} e^{x^{3}} - 8 \right) \, dx = \frac{1}{3} \left( e^{27} - e^{8} \right) - 8. \] ### Final Answer: \[ \boxed{\frac{e^{27} - e^{8}}{3}\,-\,8} \]