`intsqrt(1+2tanx(secx+tanx))dx=`

Options

A. `log_(e)|sec^(2)x+tanxsecx|+c`
B. `log_(e)|1+tanx(secx-tanx|+c`
C. `log_(e)|sinx(secx-tanx)|+c`
D. none of these

Answer

To solve the integral \( \int \sqrt{1 + 2\tan x (\sec x + \tan x)} \, dx \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ 1 + \tan^2 x = \sec^2 x \]
Thus, we can express \( 1 \) as:
\[ 1 = \sec^2 x - \tan^2 x \]

### Step 2: Substitute in the Integral
Substituting this into the integral, we have:
\[ \int \sqrt{\sec^2 x - \tan^2 x + 2\tan x (\sec x + \tan x)} \, dx \]

### Step 3: Simplify the Expression Inside the Square Root
Now, simplify the expression inside the square root:
\[ \sec^2 x - \tan^2 x + 2\tan x (\sec x + \tan x) \]
This can be rewritten as:
\[ \sec^2 x - \tan^2 x + 2\tan x \sec x + 2\tan^2 x \]
Combining like terms gives:
\[ \sec^2 x + \tan^2 x + 2\tan x \sec x \]

### Step 4: Factor the Expression
Notice that:
\[ \sec^2 x + \tan^2 x + 2\tan x \sec x = (\sec x + \tan x)^2 \]
Thus, we can rewrite our integral as:
\[ \int \sqrt{(\sec x + \tan x)^2} \, dx \]

### Step 5: Simplify the Integral
Taking the square root gives:
\[ \int |\sec x + \tan x| \, dx \]
Assuming \( \sec x + \tan x > 0 \) in the interval of integration, we can drop the absolute value:
\[ \int (\sec x + \tan x) \, dx \]

### Step 6: Integrate
The integral of \( \sec x + \tan x \) is a known result:
\[ \int \sec x \, dx = \ln |\sec x + \tan x| + C \]
Thus:
\[ \int (\sec x + \tan x) \, dx = \ln |\sec x + \tan x| + C \]

### Final Answer
Therefore, the final result is:
\[ \int \sqrt{1 + 2\tan x (\sec x + \tan x)} \, dx = \ln |\sec x + \tan x| + C \]