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The given problem is to evaluate the integral:

\[
\int \frac{3 \cos(x)}{3 + \sin(x)} \, dx
\]

Let's solve this step by step.

### Step 1: Substitution
Use the substitution:
\[
u = 3 + \sin(x)
\]
Then, differentiate \(u\) with respect to \(x\):
\[
\frac{du}{dx} = \cos(x) \implies du = \cos(x) \, dx
\]

### Step 2: Rewrite the Integral
Substituting \(u\) into the integral:
\[
\int \frac{3 \cos(x)}{3 + \sin(x)} \, dx = \int \frac{3}{u} \, du
\]

### Step 3: Integrate
The integral simplifies to:
\[
\int \frac{3}{u} \, du = 3 \ln |u| + C
\]

### Step 4: Substitute Back
Replace \(u\) with the original expression:
\[
3 \ln |3 + \sin(x)| + C
\]

Thus, the evaluated integral is:

\[
\int \frac{3 \cos(x)}{3 + \sin(x)} \, dx = 3 \ln |3 + \sin(x)| + C
\]

Would you like a detailed explanation of any specific step, or do you have any further questions?

### Related Questions:
1. How would you solve the integral \(\int \frac{\cos(x)}{3 + \sin(x)} \, dx\)?
2. What is the integral of \(\frac{1}{3 + \sin(x)}\)?
3. How can substitution be used in solving integrals involving trigonometric functions?
4. What are common techniques for integrating rational functions involving trigonometric terms?
5. How does substitution simplify the integration process?

**Tip:** When integrating, always consider using substitution if you notice a function and its derivative within the integrand.

Evaluate the integral \(\int \frac{3 \cos(x)}{3 + \sin(x)} \, dx\)

Learn how to evaluate the integral \(\int \frac{3 \cos(x)}{3 + \sin(x)} \, dx\) using trigonometric substitution. This solution involves the substitution method, leading to an integral involving natural logarithms, suitable for students in Grades 11-12 or college-level calculus courses.

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