Evaluate the following integrals:
`int (1)/(sin^(2)x cos^(2)x) dx`

Answer

To evaluate the integral

\[ \int \frac{1}{\sin^2 x \cos^2 x} \, dx, \]

we can follow these steps:

### Step 1: Rewrite the integrand
We can express the integrand in a more manageable form. We know that \( \sin^2 x + \cos^2 x = 1 \). Thus, we can write:

\[ \frac{1}{\sin^2 x \cos^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x}. \]

### Step 2: Split the integral
Now we can split the integral into two separate parts:

\[ \int \frac{1}{\sin^2 x \cos^2 x} \, dx = \int \frac{\sin^2 x}{\sin^2 x \cos^2 x} \, dx + \int \frac{\cos^2 x}{\sin^2 x \cos^2 x} \, dx. \]

This simplifies to:

\[ \int \frac{1}{\cos^2 x} \, dx + \int \frac{1}{\sin^2 x} \, dx. \]

### Step 3: Simplify the integrals
Now, we can rewrite these integrals using trigonometric identities:

1. The integral \( \int \frac{1}{\cos^2 x} \, dx \) is equal to \( \int \sec^2 x \, dx \).
2. The integral \( \int \frac{1}{\sin^2 x} \, dx \) is equal to \( \int \csc^2 x \, dx \).

### Step 4: Evaluate the integrals
Now we can evaluate each integral:

1. The integral of \( \sec^2 x \) is \( \tan x + C_1 \).
2. The integral of \( \csc^2 x \) is \( -\cot x + C_2 \).

Combining these results, we get:

\[ \int \sec^2 x \, dx + \int \csc^2 x \, dx = \tan x - \cot x + C, \]

where \( C = C_1 + C_2 \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:

\[ \int \frac{1}{\sin^2 x \cos^2 x} \, dx = \tan x - \cot x + C. \]

---