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

Options

A. `tan x + cot x + C`
B. `tan x + cosec x + C`
C. `-tan x + cot x + C`
D. `tan x + sec x + C`

Answer

To solve the integral

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

we can break it down step by step.

### Step 1: Rewrite the Integral
We start with the given integral:

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

We can separate the terms in the numerator:

\[ \int \left( \frac{\sin^2 x}{\sin^2 x \cos^2 x} - \frac{\cos^2 x}{\sin^2 x \cos^2 x} \right) \, dx. \]

### Step 2: Simplify Each Term
This simplifies to:

\[ \int \left( \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \right) \, dx. \]

### Step 3: Rewrite Using Trigonometric Functions
We can express the integrals in terms of secant and cosecant:

\[ \int \sec^2 x \, dx - \int \csc^2 x \, dx. \]

### Step 4: Integrate Each Term
Now we can integrate each term separately:

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

Thus, we have:

\[ \tan x - (-\cot x) = \tan x + \cot x. \]

### Step 5: Add the Constant of Integration
Finally, we add the constant of integration \(C\):

\[ \tan x + \cot x + C. \]

### Final Answer
The final result of the integral is:

\[ \tan x + \cot x + C. \]

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