To solve the integral of \( \cos^3(3x) \) using trigonometric substitution, we follow these steps:

### Step 1: Express the integrand in a more manageable form
Using the identity:
\[
\cos^3(\theta) = \cos(\theta) \cdot \cos^2(\theta)
\]
and recognizing that \( \cos^2(\theta) \) can be rewritten using the Pythagorean identity:
\[
\cos^2(\theta) = 1 - \sin^2(\theta),
\]
we express \( \cos^3(3x) \) as:
\[
\cos^3(3x) = \cos(3x) \cdot (1 - \sin^2(3x)).
\]

Thus, the integral becomes:
\[
\int \cos^3(3x) \, dx = \int \cos(3x) \cdot (1 - \sin^2(3x)) \, dx.
\]

### Step 2: Break the integral into two simpler integrals
We can now split this into two terms:
\[
\int \cos(3x)(1 - \sin^2(3x)) \, dx = \int \cos(3x) \, dx - \int \cos(3x) \sin^2(3x) \, dx.
\]

### Step 3: Solve the first integral
The first integral is straightforward:
\[
\int \cos(3x) \, dx = \frac{1}{3} \sin(3x) + C_1.
\]

### Step 4: Use substitution for the second integral
For the second integral \( \int \cos(3x) \sin^2(3x) \, dx \), use the substitution:
\[
u = \sin(3x), \quad du = 3 \cos(3x) \, dx \quad \text{or} \quad \frac{du}{3} = \cos(3x) \, dx.
\]

Substituting into the integral:
\[
\int \cos(3x) \sin^2(3x) \, dx = \frac{1}{3} \int u^2 \, du.
\]

### Step 5: Solve the simplified integral
Now, integrate \( u^2 \):
\[
\int u^2 \, du = \frac{u^3}{3} = \frac{\sin^3(3x)}{3}.
\]

Thus, the second integral becomes:
\[
\frac{1}{3} \int u^2 \, du = \frac{1}{9} \sin^3(3x) + C_2.
\]

### Step 6: Combine the results
Now, combine the results from both integrals:
\[
\int \cos^3(3x) \, dx = \frac{1}{3} \sin(3x) - \frac{1}{9} \sin^3(3x) + C,
\]
where \( C = C_1 + C_2 \) is the constant of integration.

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

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. What is the integral of \( \cos^3(x) \)?
2. How do you integrate \( \cos^4(x) \) using trigonometric identities?
3. What is the result of integrating \( \sin^3(2x) \)?
4. How would you approach the integral of \( \sin^2(x) \cos(x) \)?
5. What is the integral of \( \cos(2x) \sin(x) \)?

**Tip**: Always look for ways to break down powers of trigonometric functions using identities like \( \cos^2(x) = 1 - \sin^2(x) \) to simplify the integral.

Learn how to solve the integral of cos^3(3x) using trigonometric substitution. This step-by-step guide covers breaking down the integral, using Pythagorean identities, and applying substitution to simplify complex integrals. Suitable for advanced high school and college-level students.

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