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The integral you provided is:

\[
\int (6x - 5)^3 \, dx
\]

To solve this, we can use substitution. Let’s proceed step by step:

1. **Substitution**:
   Let \( u = 6x - 5 \). Then, the derivative of \( u \) with respect to \( x \) is:
   \[
   \frac{du}{dx} = 6 \quad \Rightarrow \quad dx = \frac{du}{6}
   \]

2. **Rewrite the integral**:
   Substituting into the original integral, we have:
   \[
   \int (6x - 5)^3 \, dx = \int u^3 \cdot \frac{du}{6}
   \]
   This simplifies to:
   \[
   \frac{1}{6} \int u^3 \, du
   \]

3. **Integrate**:
   The integral of \( u^3 \) is:
   \[
   \frac{u^4}{4}
   \]
   So the expression becomes:
   \[
   \frac{1}{6} \cdot \frac{u^4}{4} = \frac{u^4}{24}
   \]

4. **Substitute back for \( u \)**:
   Since \( u = 6x - 5 \), we substitute back to get:
   \[
   \frac{(6x - 5)^4}{24} + C
   \]
   where \( C \) is the constant of integration.

Thus, the final solution is:

\[
\frac{(6x - 5)^4}{24} + C
\]

Would you like more details on any of the steps?

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Here are five related questions to deepen your understanding:
1. How would the integral change if the exponent were different, say \( (6x - 5)^n \)?
2. Can you apply integration by parts to solve this problem?
3. How would you handle this integral if it were a definite integral with specific limits?
4. What are the general rules for using substitution in integrals?
5. How would this integral change if the expression inside the parentheses were more complex?

**Tip**: Always check if substitution simplifies an integral before trying more complex methods like integration by parts.

Solve the integral ∫ (6x - 5)^3 dx using substitution. Learn how to apply the substitution method, including key steps for simplifying the expression and integrating powers of u. Suitable for undergraduate calculus students.

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