`int(dx)/( sin^(2)x cos^(2)x)` is equal to

Options

A. `tanx + cot x+C`
B. `( tanx + cot x )^(2) +C`
C. `tanx - cot x +C`
D. `( tan x -cot x) ^(2) + C`

Answer

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

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \frac{dx}{\sin^2 x \cos^2 x} \]
Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can express 1 in terms of \( \sin^2 x \) and \( \cos^2 x \):
\[ 1 = \sin^2 x + \cos^2 x \]
Thus, we can rewrite the integral as:
\[ \int \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} \, dx \]

### Step 2: Split the Integral
Now, we can split the integral into two parts:
\[ \int \left( \frac{\sin^2 x}{\sin^2 x \cos^2 x} + \frac{\cos^2 x}{\sin^2 x \cos^2 x} \right) dx \]
This simplifies to:
\[ \int \left( \frac{1}{\cos^2 x} + \frac{1}{\sin^2 x} \right) dx \]

### Step 3: Use Trigonometric Identities
We know that:
\[ \frac{1}{\cos^2 x} = \sec^2 x \quad \text{and} \quad \frac{1}{\sin^2 x} = \csc^2 x \]
Thus, the integral now becomes:
\[ \int \sec^2 x \, dx + \int \csc^2 x \, dx \]

### Step 4: Integrate Each Term
The integrals of \( \sec^2 x \) and \( \csc^2 x \) are well-known:
\[ \int \sec^2 x \, dx = \tan x + C_1 \]
\[ \int \csc^2 x \, dx = -\cot x + C_2 \]
Combining these results, we have:
\[ \tan x - \cot x + C \]
where \( C = C_1 + C_2 \) is the constant of integration.

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

---