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

Options

A. `x+secx+tanx`
B. `2(secx+tanx)-x`
C. `2(secx-tanx)+x`
D. `2(secx-tanx)+x`

Answer

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

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

### Step 2: Substitute \( \tan^2 x \)
Using the identity \( \tan^2 x = \sec^2 x - 1 \), we can rewrite the integral:
\[ \sec^2 x + 2 \sec x \tan x + \tan^2 x = \sec^2 x + 2 \sec x \tan x + (\sec^2 x - 1) \]
This simplifies to:
\[ 2 \sec^2 x + 2 \sec x \tan x - 1 \]

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

### 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: Integrate each term
1. The integral of \( 2 \sec^2 x \):
\[ \int 2 \sec^2 x \, dx = 2 \tan x \]
2. The integral of \( 2 \sec x \tan x \):
\[ \int 2 \sec x \tan x \, dx = 2 \sec x \]
3. The integral of \( 1 \):
\[ \int 1 \, dx = x \]

### Step 6: Combine the results
Now, we combine all the results:
\[ 2 \tan x + 2 \sec x - x + C \]
where \( C \) is the constant of integration.

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