`int(tan^(2)x-cot^(2)x)dx=`

Options

A. `tanx+cotx+c`
B. `tanx-cotx+c`
C. `-tanx-cotx+4x+c`
D. `cot2x+cotx+c`

Answer

To solve the integral \( \int (\tan^2 x - \cot^2 x) \, dx \), we can follow these steps:

### Step 1: Rewrite the integrand
We start with the expression \( \tan^2 x - \cot^2 x \). We can use the identities for tangent and cotangent:
\[ \tan^2 x = \sec^2 x - 1 \quad \text{and} \quad \cot^2 x = \csc^2 x - 1 \]
Substituting these identities into the integral, we have:
\[ \tan^2 x - \cot^2 x = (\sec^2 x - 1) - (\csc^2 x - 1) \]
This simplifies to:
\[ \tan^2 x - \cot^2 x = \sec^2 x - \csc^2 x \]

### Step 2: Set up the integral
Now we can rewrite the integral:
\[ \int (\tan^2 x - \cot^2 x) \, dx = \int (\sec^2 x - \csc^2 x) \, dx \]

### Step 3: Separate the integrals
We can separate the integral into two parts:
\[ \int \sec^2 x \, dx - \int \csc^2 x \, dx \]

### Step 4: Integrate each part
We know the integrals of the secant and cosecant functions:
\[ \int \sec^2 x \, dx = \tan x + C_1 \]
\[ \int \csc^2 x \, dx = -\cot x + C_2 \]
Thus, we have:
\[ \int \sec^2 x \, dx - \int \csc^2 x \, dx = \tan x - (-\cot x) + C \]
This simplifies to:
\[ \tan x + \cot x + C \]

### Final Answer
Therefore, the final result of the integral is:
\[ \int (\tan^2 x - \cot^2 x) \, dx = \tan x + \cot x + C \]