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.