solve `intcot(x/3)dx`

Answer

To solve the integral \( \int \cot\left(\frac{x}{3}\right) dx \), we can follow these steps:

### Step 1: Substitution
Let \( t = \frac{x}{3} \). Then, we differentiate both sides to find \( dx \):
\[ dt = \frac{1}{3} dx \quad \Rightarrow \quad dx = 3 dt \]

### Step 2: Rewrite the Integral
Substituting \( t \) into the integral, we have:
\[ \int \cot\left(\frac{x}{3}\right) dx = \int \cot(t) \cdot 3 dt = 3 \int \cot(t) dt \]

### Step 3: Integrate \( \cot(t) \)
The integral of \( \cot(t) \) is known:
\[ \int \cot(t) dt = \ln|\sin(t)| + C \]
Thus,
\[ 3 \int \cot(t) dt = 3 \left( \ln|\sin(t)| + C \right) = 3 \ln|\sin(t)| + 3C \]

### Step 4: Substitute Back for \( t \)
Now, substituting back \( t = \frac{x}{3} \):
\[ 3 \ln|\sin(t)| + 3C = 3 \ln|\sin\left(\frac{x}{3}\right)| + C' \]
where \( C' = 3C \) is just a constant of integration.

### Final Answer
Thus, the final result of the integral is:
\[ \int \cot\left(\frac{x}{3}\right) dx = 3 \ln|\sin\left(\frac{x}{3}\right)| + C \]

---