`intcos(5+6x)dx`

Answer

To solve the integral \(\int \cos(5 + 6x) \, dx\), we can use a substitution method. Here’s a step-by-step solution:

### Step 1: Identify the substitution
Let \( u = 5 + 6x \). Then, we need to find \( du \).

### Step 2: Differentiate the substitution
Differentiating \( u \) with respect to \( x \):
\[ du = 6 \, dx \quad \Rightarrow \quad dx = \frac{du}{6} \]

### Step 3: Substitute in the integral
Now, substitute \( u \) and \( dx \) into the integral:
\[ \int \cos(5 + 6x) \, dx = \int \cos(u) \cdot \frac{du}{6} \]

### Step 4: Factor out the constant
We can factor out the constant \( \frac{1}{6} \):
\[ = \frac{1}{6} \int \cos(u) \, du \]

### Step 5: Integrate
The integral of \( \cos(u) \) is \( \sin(u) \):
\[ = \frac{1}{6} \sin(u) + C \]

### Step 6: Substitute back for \( u \)
Now, substitute back \( u = 5 + 6x \):
\[ = \frac{1}{6} \sin(5 + 6x) + C \]

### Final Answer
Thus, the final answer is:
\[ \int \cos(5 + 6x) \, dx = \frac{1}{6} \sin(5 + 6x) + C \]