Evaluate `int(secx+tanx)^(2)dx`

Explanation

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

### Step 1: Expand the Expression
First, we expand the integrand:
\[ (\sec x + \tan x)^2 = \sec^2 x + 2 \sec x \tan x + \tan^2 x \]

### Step 2: Substitute for \(\tan^2 x\)
Using the identity \( \tan^2 x = \sec^2 x - 1 \), we can substitute this into our expression:
\[ \sec^2 x + 2 \sec x \tan x + \tan^2 x = \sec^2 x + 2 \sec x \tan x + (\sec^2 x - 1) \]
This simplifies to:
\[ 2 \sec^2 x + 2 \sec x \tan x - 1 \]

### Step 3: Rewrite the Integral
Now we rewrite the integral:
\[ \int (\sec x + \tan x)^2 \, dx = \int (2 \sec^2 x + 2 \sec x \tan x - 1) \, dx \]

### Step 4: Split the Integral
We can split the integral into three separate integrals:
\[ \int (2 \sec^2 x) \, dx + \int (2 \sec x \tan x) \, dx - \int 1 \, dx \]

### Step 5: Evaluate Each Integral
1. The integral of \(2 \sec^2 x\) is:
\[ 2 \tan x \]
2. The integral of \(2 \sec x \tan x\) is:
\[ 2 \sec x \]
3. The integral of \(1\) is:
\[ x \]

### Step 6: Combine the Results
Combining all the results, we have:
\[ \int (\sec x + \tan x)^2 \, dx = 2 \tan x + 2 \sec x - x + C \]

### Final Answer
Thus, the final answer is:
\[ \int (\sec x + \tan x)^2 \, dx = 2 \sec x + 2 \tan x - x + C \]