`int (1)/(sin^(2) x cos^(2)x) dx=?`

Answer

To solve the integral \( \int \frac{1}{\sin^2 x \cos^2 x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \frac{1}{\sin^2 x \cos^2 x} \, dx \]
We can rewrite this using the identity \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x) \):
\[ \int \frac{4}{\sin^2(2x)} \, dx \]

### Step 2: Use the Cosecant Function
We know that \( \frac{1}{\sin^2(2x)} = \csc^2(2x) \). Thus, we can rewrite the integral as:
\[ 4 \int \csc^2(2x) \, dx \]

### Step 3: Integrate
The integral of \( \csc^2(kx) \) is \( -\frac{1}{k} \cot(kx) + C \). Here, \( k = 2 \):
\[ 4 \int \csc^2(2x) \, dx = 4 \left(-\frac{1}{2} \cot(2x) + C\right) = -2 \cot(2x) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1}{\sin^2 x \cos^2 x} \, dx = -2 \cot(2x) + C \]

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