`int{(1-tan((x)/(2)))/(1+tan((x)/(2)))}dx=?`

Options

A. `2log|"sec"(x)/(2)|+C`
B. `2log|"cos"(x)/(2)|+C`
C. `2log|sec((pi)/(4)-(x)/(2))|+C`
D. `2log|cos((pi)/(4)-(x)/(2))|+C`

Explanation

To solve the integral \( \int \frac{1 - \tan\left(\frac{x}{2}\right)}{1 + \tan\left(\frac{x}{2}\right)} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can rewrite the expression using the identity for tangent subtraction:
\[ \frac{1 - \tan\left(\frac{x}{2}\right)}{1 + \tan\left(\frac{x}{2}\right)} = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \]
Thus, the integral becomes:
\[ \int \tan\left(\frac{\pi}{4} - \frac{x}{2}\right) \, dx \]

### Step 2: Substitution
Let:
\[ t = \frac{\pi}{4} - \frac{x}{2} \]
Then, differentiating both sides gives:
\[ dt = -\frac{1}{2} dx \quad \Rightarrow \quad dx = -2 dt \]

### Step 3: Substitute in the Integral
Substituting \( t \) into the integral:
\[ \int \tan(t) (-2 dt) = -2 \int \tan(t) \, dt \]

### Step 4: Integrate
The integral of \( \tan(t) \) is:
\[ \int \tan(t) \, dt = -\log|\cos(t)| + C \]
Thus:
\[ -2 \int \tan(t) \, dt = -2 \left(-\log|\cos(t)|\right) + C = 2 \log|\cos(t)| + C \]

### Step 5: Back Substitute
Now substitute back \( t = \frac{\pi}{4} - \frac{x}{2} \):
\[ 2 \log|\cos\left(\frac{\pi}{4} - \frac{x}{2}\right)| + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1 - \tan\left(\frac{x}{2}\right)}{1 + \tan\left(\frac{x}{2}\right)} \, dx = 2 \log\left|\cos\left(\frac{\pi}{4} - \frac{x}{2}\right)\right| + C \]