To solve the integral of \(\frac{x^6}{7} \, dx\), follow these steps:

### Step-by-Step Solution:

1. Factor out the constant \(\frac{1}{7}\) from the integral:
   \[
   \int \frac{x^6}{7} \, dx = \frac{1}{7} \int x^6 \, dx
   \]

2. Now, integrate \(x^6\) using the power rule:
   \[
   \int x^6 \, dx = \frac{x^{6+1}}{6+1} = \frac{x^7}{7}
   \]

3. Substitute this result back into the expression:
   \[
   \frac{1}{7} \times \frac{x^7}{7} = \frac{x^7}{49}
   \]

4. Don’t forget the constant of integration \(C\), since this is an indefinite integral.

### Final Answer:
\[
\int \frac{x^6}{7} \, dx = \frac{x^7}{49} + C
\]

Would you like more details on any step or have any further questions?

Here are 5 related questions to explore:

1. What is the integral of \(\frac{x^5}{7}\)?
2. How do you solve the integral of \(\frac{x^n}{a}\)?
3. Can you explain the power rule for integration in more detail?
4. How do constants affect integration?
5. How do definite integrals differ from indefinite integrals?

**Tip**: Always remember to add the constant \(C\) when solving indefinite integrals!

Solve the integral of \frac{x^6}{7} step-by-step using the power rule. This calculus problem explains how to integrate functions involving powers of x, factoring constants, and applying the indefinite integral with the constant of integration. Suitable for students in Grades 11-12.

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