To evaluate the integral of \( \sin^5(9x) \, dx \), we can use trigonometric identities and substitution. Here's the step-by-step process:

### Step 1: Rewrite \( \sin^5(9x) \)
We can express \( \sin^5(9x) \) as:
\[
\sin^5(9x) = \sin^4(9x) \cdot \sin(9x)
\]
Now, use the identity \( \sin^2(\theta) = 1 - \cos^2(\theta) \):
\[
\sin^4(9x) = (\sin^2(9x))^2 = (1 - \cos^2(9x))^2
\]
Thus, we rewrite:
\[
\sin^5(9x) = (1 - \cos^2(9x))^2 \cdot \sin(9x)
\]

### Step 2: Substitution
Let:
\[
u = \cos(9x), \quad \frac{du}{dx} = -9 \sin(9x)
\]
So, the integral becomes:
\[
\int \sin^5(9x) \, dx = \int (1 - u^2)^2 (-\frac{1}{9}) \, du
\]
Simplify:
\[
-\frac{1}{9} \int (1 - u^2)^2 \, du
\]

### Step 3: Expand and Integrate
Expand \( (1 - u^2)^2 \):
\[
(1 - u^2)^2 = 1 - 2u^2 + u^4
\]
Now, integrate each term:
\[
-\frac{1}{9} \left( \int 1 \, du - 2 \int u^2 \, du + \int u^4 \, du \right)
\]
The integrals are straightforward:
\[
\int 1 \, du = u, \quad \int u^2 \, du = \frac{u^3}{3}, \quad \int u^4 \, du = \frac{u^5}{5}
\]

### Step 4: Combine the Results
Now substitute back the results of the integrals:
\[
-\frac{1}{9} \left( u - 2 \cdot \frac{u^3}{3} + \frac{u^5}{5} \right) + C
\]
Substitute \( u = \cos(9x) \):
\[
-\frac{1}{9} \left( \cos(9x) - \frac{2}{3} \cos^3(9x) + \frac{1}{5} \cos^5(9x) \right) + C
\]

### Final Answer:
\[
\int \sin^5(9x) \, dx = -\frac{1}{9} \left( \cos(9x) - \frac{2}{3} \cos^3(9x) + \frac{1}{5} \cos^5(9x) \right) + C
\]

Would you like further details or clarifications?

Here are 5 related questions you might consider:
1. How would the method change if the power were even, such as \( \sin^6(9x) \)?
2. What is the integral of \( \cos^5(9x) \, dx \)?
3. How does the trigonometric identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) apply to integral problems like this?
4. What happens if we integrate \( \sin^n(9x) \, dx \) for a general \( n \)?
5. How would substitution change if we dealt with \( \sin^5(ax) \, dx \) instead?

**Tip:** For odd powers of sine or cosine, splitting off one sine or cosine term and using trigonometric identities often simplifies the integration process.

Explore how to evaluate the integral of sin^5(9x) using trigonometric identities and substitution. This detailed step-by-step guide covers important concepts such as substitution and expansion, suitable for advanced high school or undergraduate calculus students.

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