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

Options

A. `tanx+cotx`
B. `tanx-cotx`
C. `secx+tanx`
D. `2cot2x+c`

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 can rewrite the integral as:
\[ \int \frac{1}{\sin^2 x \cos^2 x} \, dx = \int \frac{1}{\sin^2 x} \cdot \frac{1}{\cos^2 x} \, dx \]

### Step 2: Use Trigonometric Identities
Recall that:
\[ \frac{1}{\sin^2 x} = \csc^2 x \quad \text{and} \quad \frac{1}{\cos^2 x} = \sec^2 x \]
Thus, we can rewrite the integral as:
\[ \int \csc^2 x \sec^2 x \, dx \]

### Step 3: Split the Integral
We can split the integral into two parts:
\[ \int \csc^2 x \sec^2 x \, dx = \int \csc^2 x \, dx + \int \sec^2 x \, dx \]

### Step 4: Integrate Each Part
Now we can integrate each part separately:
1. The integral of \( \csc^2 x \) is:
\[ \int \csc^2 x \, dx = -\cot x + C_1 \]
2. The integral of \( \sec^2 x \) is:
\[ \int \sec^2 x \, dx = \tan x + C_2 \]

### Step 5: Combine the Results
Combining the results from both integrals, we have:
\[ \int \csc^2 x \sec^2 x \, dx = -\cot x + \tan x + C \]
where \( C = C_1 + C_2 \) is the constant of integration.

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

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