1. **State the problem:** Calculate the integral $$\int \left(x^2 + \frac{1}{6}x - 9x^2\right) \, dx$$ and explain the steps clearly. 2. **Simplify the integrand:** Combine like terms inside the integral. $$ x^2 - 9x^2 + \frac{1}{6}x = (1 - 9)x^2 + \frac{1}{6}x = -8x^2 + \frac{1}{6}x $$ 3. **Rewrite the integral:** $$ \int \left(-8x^2 + \frac{1}{6}x\right) dx $$ 4. **Integrate term by term:** Use the power rule for integration: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ - Integral of $$-8x^2$$ is $$-8 \times \frac{x^3}{3} = -\frac{8}{3}x^3$$. - Integral of $$\frac{1}{6}x$$ is $$\frac{1}{6} \times \frac{x^2}{2} = \frac{1}{12}x^2$$. 5. **Write the final answer:** $$ \int \left(x^2 + \frac{1}{6}x - 9x^2\right) dx = -\frac{8}{3}x^3 + \frac{1}{12}x^2 + C $$ where $$C$$ is the constant of integration.