`int " cosec"^(4) 2x dx`

Answer

To solve the integral \( \int \csc^4(2x) \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \csc^4(2x) \, dx \]
We can express \( \csc^4(2x) \) as \( \csc^2(2x) \cdot \csc^2(2x) \):
\[ \int \csc^4(2x) \, dx = \int \csc^2(2x) \cdot \csc^2(2x) \, dx \]

### Step 2: Use the Identity for Cosecant
We know that:
\[ \csc^2(2x) = 1 + \cot^2(2x) \]
Thus, we can rewrite the integral:
\[ \int \csc^4(2x) \, dx = \int \csc^2(2x) (1 + \cot^2(2x)) \, dx \]
This expands to:
\[ \int \csc^2(2x) \, dx + \int \csc^2(2x) \cot^2(2x) \, dx \]

### Step 3: Solve the First Integral
The integral \( \int \csc^2(2x) \, dx \) is straightforward:
\[ \int \csc^2(2x) \, dx = -\frac{1}{2} \cot(2x) + C_1 \]

### Step 4: Solve the Second Integral
Now we focus on the second integral:
\[ \int \csc^2(2x) \cot^2(2x) \, dx \]
We can use substitution. Let:
\[ t = \cot(2x) \implies dt = -2 \csc^2(2x) \, dx \implies dx = -\frac{dt}{2 \csc^2(2x)} \]
Substituting this into the integral gives:
\[ \int \csc^2(2x) \cot^2(2x) \, dx = \int t^2 \left(-\frac{dt}{2}\right) = -\frac{1}{2} \int t^2 \, dt \]
Calculating this integral:
\[ -\frac{1}{2} \cdot \frac{t^3}{3} = -\frac{t^3}{6} \]

### Step 5: Substitute Back
Substituting back \( t = \cot(2x) \):
\[ -\frac{1}{6} \cot^3(2x) \]

### Step 6: Combine Results
Now we combine the results from the two integrals:
\[ \int \csc^4(2x) \, dx = -\frac{1}{2} \cot(2x) - \frac{1}{6} \cot^3(2x) + C \]

### Final Answer
Thus, the final result is:
\[ \int \csc^4(2x) \, dx = -\frac{1}{2} \cot(2x) - \frac{1}{6} \cot^3(2x) + C \]