5. Evaluate \( \int_{-1}^{3}\left(x^{2}-3\right)\left(x^{3}-9 x-6\right)^{3} d x \)

To evaluate the integral \[ I = \int_{-1}^{3}(x^{2}-3)(x^{3}-9x-6)^{3} \, dx, \] we can start by considering the properties of the integrand, particularly focus on symmetry. Notice that the limits of integration are from \(-1\) to \(3\). Let's first inspect the polynomial \(x^{3} - 9x - 6\) at the endpoints of the integration. Calculate the value of the polynomial at the limits: - For \(x = -1\): \[ x^{3} - 9x - 6 = (-1)^{3} - 9(-1) - 6 = -1 + 9 - 6 = 2. \] - For \(x = 3\): \[ x^{3} - 9x - 6 = (3)^{3} - 9(3) - 6 = 27 - 27 - 6 = -6. \] Next, observe the behavior of \(x^{2} - 3\) over our range of integration. The function \(x^{2} - 3\) is negative when \(-1 < x < \sqrt{3}\) and positive when \(\sqrt{3} < x < 3\). Next, we compute the integral over the specified bounds. Note that the function \(x^3 - 9x - 6\) is a continuous function and we can check if it has roots within the range \(-1\) to \(3\). The roots will help determine the sign of \((x^3 - 9x - 6)^3\). A critical insight is to find out if there is a transformation we can apply that reflects the polynomial about some midpoint, potentially helping to simplify the computation. Let’s define a substitution \(u = x - 1\), which will shift our limits of integration from \([-1, 3]\) to \([-2, 2]\). Substitute into our integral: \[ I = \int_{-2}^{2}((u+1)^{2}-3)((u+1)^{3}-9(u+1)-6)^{3} \, du. \] This form can become quite complex. However, instead, we can look at the original polynomial used, noticing that it is an odd polynomial when correctly formatted. There’s a symmetry between the integral bounds (the center of integration is at \(1\), which lies in the domain, lending to potential cancellation of positive and negative areas. The integrand \((x^{2}-3)(x^{3}-9x-6)^{3}\) has odd symmetry around the midpoint \(x=1\). In particular, we can show: \[ I = f(-x) + f(x) \quad \text{(odd functions) implies } I \text{ evaluates to } 0. \] Thus, we conclude that \[ \int_{-1}^{3}(x^{2}-3)(x^{3}-9x-6)^{3} \,dx = 0. \] The final answer is: \[ \boxed{0}. \]