Use the Fundamental Theorem of Calculus to evaluate the following definite integral. \( \int_{-1}^{5}\left(5 x^{3}+2\right) d x \) \( \int_{-1}^{5}\left(5 x^{3}+2\right) d x=\square \) (Type an exact answer.)

To evaluate the definite integral \( \int_{-1}^{5}\left(5 x^{3}+2\right) d x \) using the Fundamental Theorem of Calculus, we first need to find an antiderivative of the function \( f(x) = 5x^3 + 2 \). The antiderivative is: \[ F(x) = \frac{5}{4}x^4 + 2x + C \] Now, we apply the limits of integration from -1 to 5: \[ \int_{-1}^{5}\left(5 x^{3}+2\right) d x = F(5) - F(-1) \] Calculating \( F(5) \): \[ F(5) = \frac{5}{4}(5)^4 + 2(5) = \frac{5}{4}(625) + 10 = \frac{3125}{4} + 10 = \frac{3125}{4} + \frac{40}{4} = \frac{3165}{4} \] Calculating \( F(-1) \): \[ F(-1) = \frac{5}{4}(-1)^4 + 2(-1) = \frac{5}{4}(1) - 2 = \frac{5}{4} - \frac{8}{4} = -\frac{3}{4} \] Now, substitute these values back in: \[ \int_{-1}^{5}\left(5 x^{3}+2\right) d x = \frac{3165}{4} - \left(-\frac{3}{4}\right) = \frac{3165}{4} + \frac{3}{4} = \frac{3168}{4} = 792 \] Thus, the exact answer is: \[ \int_{-1}^{5}\left(5 x^{3}+2\right) d x = 792 \]