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To solve the integral

\[
\int x \cot^2 (4x) \, dx,
\]

we can approach it as follows:

1. **Rewrite** the integrand by using trigonometric identities. Since \( \cot^2 u = \csc^2 u - 1 \), we can express \( \cot^2 (4x) \) in terms of \( \csc^2 (4x) \):
   \[
   \int x \cot^2 (4x) \, dx = \int x (\csc^2 (4x) - 1) \, dx.
   \]
   Expanding this, we get:
   \[
   = \int x \csc^2 (4x) \, dx - \int x \, dx.
   \]

2. **Evaluate** each integral separately.

   - For the first integral, \(\int x \csc^2 (4x) \, dx\), we can use **integration by parts** where we set:
     - \(u = x\), so \(du = dx\),
     - \(dv = \csc^2 (4x) \, dx\), so \(v = -\frac{1}{4} \cot (4x)\) (since \(\int \csc^2 u \, du = -\cot u\)).

     Using integration by parts, we get:
     \[
     \int x \csc^2 (4x) \, dx = -\frac{x}{4} \cot (4x) + \frac{1}{4} \int \cot (4x) \, dx.
     \]

   - For \(\int \cot (4x) \, dx\), we know:
     \[
     \int \cot (4x) \, dx = \frac{1}{4} \ln |\sin (4x)|.
     \]
     Substituting this back, we have:
     \[
     \int x \csc^2 (4x) \, dx = -\frac{x}{4} \cot (4x) + \frac{1}{16} \ln |\sin (4x)|.
     \]

   - For the second integral, \(\int x \, dx\):
     \[
     \int x \, dx = \frac{x^2}{2}.
     \]

3. **Combine** all parts:
   \[
   \int x \cot^2 (4x) \, dx = -\frac{x}{4} \cot (4x) + \frac{1}{16} \ln |\sin (4x)| - \frac{x^2}{2} + C.
   \]

The answer that matches this expression is option **C**:

\[
-\frac{1}{4} x \cot 4x - \frac{1}{16} \ln |\sin 4x| + C.
\]

Would you like further details on any part of this solution? Here are some questions to deepen understanding:

1. Why does integration by parts work well for this integral?
2. How can the identity \(\cot^2 u = \csc^2 u - 1\) simplify trigonometric integrals?
3. How would this integral change if \(4x\) were replaced with a different coefficient?
4. Can you explain the substitution in the integral of \(\cot(4x)\)?
5. What would happen if we tried a u-substitution instead?

**Tip:** Always check for simplifications with trigonometric identities when handling integrals involving powers of trig functions.

Learn how to solve the integral ∫ x cot²(4x) dx by using trigonometric identities and integration by parts. This solution includes detailed steps with trigonometric simplifications and integration techniques, suitable for college-level calculus students.

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