To solve the integral $\int -3x^5 \, dx$, we can use the power rule for integration, which states:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{where} \, n \neq -1$

Now apply the rule to the given function:

1. Factor out the constant $-3$: $\int -3x^5 \, dx = -3 \int x^5 \, dx$

2. Apply the power rule: $-3 \cdot \left( \frac{x^{5+1}}{5+1} \right) + C = -3 \cdot \left( \frac{x^6}{6} \right) + C$

3. Simplify: $\frac{-3x^6}{6} + C = \frac{-x^6}{2} + C$

So, the solution is: $\int -3x^5 \, dx = \frac{-x^6}{2} + C$

Would you like further explanation or have any questions?

Here are 5 related questions:

1. What is the general formula for the power rule of integration?
2. How would you integrate $\int 5x^4 \, dx$?
3. How do you handle the integration of negative powers, such as $\int x^{-2} \, dx$?
4. What is the constant of integration and why is it important?
5. How would you verify the result of this integral by differentiation?

Tip: Always remember to add the constant $C$ when performing indefinite integrals!