Evaluate the following integrals : `int(tanx+cotx)^2dx`

Answer

To evaluate the integral \(\int (\tan x + \cot x)^2 \, dx\), we will follow these steps:

### Step 1: Expand the integrand
We start by expanding the expression \((\tan x + \cot x)^2\):
\[ (\tan x + \cot x)^2 = \tan^2 x + \cot^2 x + 2 \tan x \cot x \]

### Step 2: Substitute \(\cot x\)
Recall that \(\cot x = \frac{1}{\tan x}\). Thus, we can rewrite the integral:
\[ \int (\tan^2 x + \cot^2 x + 2 \tan x \cot x) \, dx = \int \tan^2 x \, dx + \int \cot^2 x \, dx + \int 2 \tan x \cot x \, dx \]

### Step 3: Simplify the terms
Now, we know that:
\[ \tan^2 x = \sec^2 x - 1 \quad \text{and} \quad \cot^2 x = \csc^2 x - 1 \]
Substituting these into the integral gives:
\[ \int (\sec^2 x - 1) \, dx + \int (\csc^2 x - 1) \, dx + \int 2 \tan x \cot x \, dx \]
This simplifies to:
\[ \int \sec^2 x \, dx - \int 1 \, dx + \int \csc^2 x \, dx - \int 1 \, dx + \int 2 \, dx \]

### Step 4: Combine the integrals
Combining these integrals, we have:
\[ \int \sec^2 x \, dx + \int \csc^2 x \, dx - 2 \int 1 \, dx + \int 2 \, dx \]
This can be simplified to:
\[ \int \sec^2 x \, dx + \int \csc^2 x \, dx \]

### Step 5: Evaluate the integrals
We know the integrals:
\[ \int \sec^2 x \, dx = \tan x + C_1 \]
\[ \int \csc^2 x \, dx = -\cot x + C_2 \]
Thus, combining these results:
\[ \tan x - \cot x + C \]

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