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The given integral is:

\[
\int 29 \cos^3(84x) \, dx
\]

### Step-by-Step Solution:

1. **Factor out the constant:**
   The constant \(29\) can be factored out of the integral:
   \[
   29 \int \cos^3(84x) \, dx
   \]

2. **Use the trigonometric identity:**
   To simplify \( \cos^3(84x) \), use the identity:
   \[
   \cos^3(\theta) = \cos(\theta) \left( \frac{3}{4} + \frac{\cos(2\theta)}{4} \right)
   \]
   However, a more common approach uses the identity:
   \[
   \cos^3(\theta) = \frac{1}{4} (3\cos(\theta) + \cos(3\theta))
   \]
   Applying this identity to \( \cos^3(84x) \), we get:
   \[
   \cos^3(84x) = \frac{1}{4} (3\cos(84x) + \cos(252x))
   \]

3. **Rewrite the integral:**
   Now the integral becomes:
   \[
   29 \int \left( \frac{1}{4} \left( 3\cos(84x) + \cos(252x) \right) \right) dx
   \]
   Simplifying this:
   \[
   \frac{29}{4} \int \left( 3\cos(84x) + \cos(252x) \right) dx
   \]

4. **Integrate term by term:**
   - The integral of \( \cos(kx) \) is \( \frac{\sin(kx)}{k} \).
   
   Therefore:
   \[
   \int 3 \cos(84x) \, dx = \frac{3}{84} \sin(84x) = \frac{1}{28} \sin(84x)
   \]
   and
   \[
   \int \cos(252x) \, dx = \frac{1}{252} \sin(252x)
   \]

5. **Combine the results:**
   Now, substitute the integrated results:
   \[
   \frac{29}{4} \left( \frac{1}{28} \sin(84x) + \frac{1}{252} \sin(252x) \right)
   \]

6. **Simplify the expression:**
   Simplifying the coefficients:
   \[
   \frac{29}{4} \left( \frac{1}{28} \sin(84x) + \frac{1}{252} \sin(252x) \right)
   = \frac{29}{112} \sin(84x) + \frac{29}{1008} \sin(252x)
   \]

7. **Final result:**
   The indefinite integral is:
   \[
   \frac{29}{112} \sin(84x) + \frac{29}{1008} \sin(252x) + C
   \]
   where \( C \) is the constant of integration.

---

Would you like any further details or clarifications? Here are five related questions to consider:

1. How would you integrate powers of sine or cosine functions?
2. What other trigonometric identities can simplify integrals?
3. How do trigonometric substitution methods work in solving integrals?
4. What is the difference between definite and indefinite integrals?
5. How does integration by parts apply to trigonometric integrals?

**Tip:** Always factor out constants at the beginning of the integral to simplify your work!

Evaluate the indefinite integral ∫ 29 cos^3(84x) dx.

Learn how to evaluate the indefinite integral ∫ 29 cos^3(84x) dx using trigonometric identities and step-by-step integration. This example is suitable for students in Grades 11-12 studying trigonometric integrals.

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