1. **State the problem:** We want to evaluate the integral $$\int_{-t}^t x(t^2 - x^2) \, dx$$ using the power rule. 2. **Rewrite the integrand:** Distribute $x$ inside the parentheses: $$x(t^2 - x^2) = x t^2 - x^3$$ 3. **Split the integral:** $$\int_{-t}^t (x t^2 - x^3) \, dx = t^2 \int_{-t}^t x \, dx - \int_{-t}^t x^3 \, dx$$ 4. **Evaluate each integral separately:** - For $$\int_{-t}^t x \, dx$$, since $x$ is an odd function and the limits are symmetric about zero, the integral is zero. - For $$\int_{-t}^t x^3 \, dx$$, $x^3$ is also an odd function, so this integral is also zero. 5. **Conclusion:** Both integrals are zero, so $$\int_{-t}^t x(t^2 - x^2) \, dx = 0 - 0 = 0$$ **Final answer:** $$0$$