To solve the integral: $\int 5a^2 x^6 \, dx$

where $a$ is a constant, we proceed with the following steps:

Step 1: Apply the constant factor rule.

The constant $5a^2$ can be factored out of the integral: $5a^2 \int x^6 \, dx$

Step 2: Integrate $x^6$.

The integral of $x^n$ is given by: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $C$ is the constant of integration.

Applying this rule to $x^6$, we get: $\int x^6 \, dx = \frac{x^7}{7}$

Step 3: Multiply by the constant factor.

Now, multiply the result by $5a^2$: $5a^2 \cdot \frac{x^7}{7} = \frac{5a^2 x^7}{7}$

Final answer:

The indefinite integral is: $\int 5a^2 x^6 \, dx = \frac{5a^2 x^7}{7} + C$

Would you like more details, or do you have any questions?

Here are 5 related questions to consider:

1. How would the solution change if the exponent of $x$ was different?
2. What happens if $a$ is not a constant but a function of $x$?
3. How do you solve definite integrals involving powers of $x$?
4. What is the antiderivative of a sum of functions?
5. How does the power rule for integration extend to negative exponents?

Tip: When integrating, always factor out constants to simplify the integral.