`int((1-tanx)/(1+tanx))^(2)dx=`

Options

A. `-(tan((pi)/(4)-x)+x)+c`
B. `-(tan((pi)/(4)+x)+x)+c`
C. `tan((pi)/(4)-x)+x+c`
D. `tan((pi)/(4)+x)+x+c`

Answer

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

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \left( \frac{1 - \tan x}{1 + \tan x} \right)^2 dx \]

### Step 2: Use the Identity for Tangent
We can express \( 1 \) in terms of \( \tan \frac{\pi}{4} \):
\[ 1 = \tan \frac{\pi}{4} \]
Thus, we can rewrite the expression as:
\[ I = \int \left( \frac{\tan \frac{\pi}{4} - \tan x}{1 + \tan x} \right)^2 dx \]

### Step 3: Apply the Tangent Subtraction Formula
Using the tangent subtraction formula:
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
we can identify \( a = \frac{\pi}{4} \) and \( b = x \):
\[ I = \int \tan\left(\frac{\pi}{4} - x\right)^2 dx \]

### Step 4: Use the Identity for Tangent Squared
We can use the identity:
\[ \tan^2 \theta = \sec^2 \theta - 1 \]
Thus, we have:
\[ I = \int \left(\sec^2\left(\frac{\pi}{4} - x\right) - 1\right) dx \]

### Step 5: Split the Integral
Now we can split the integral:
\[ I = \int \sec^2\left(\frac{\pi}{4} - x\right) dx - \int 1 dx \]

### Step 6: Integrate Each Part
The integral of \( \sec^2 \theta \) is \( \tan \theta \):
\[ \int \sec^2\left(\frac{\pi}{4} - x\right) dx = -\tan\left(\frac{\pi}{4} - x\right) + C \]
And the integral of \( 1 \) is simply:
\[ \int 1 dx = x \]

### Step 7: Combine Results
Combining these results gives:
\[ I = -\tan\left(\frac{\pi}{4} - x\right) - x + C \]

### Step 8: Simplify
Since \( \tan\left(\frac{\pi}{4} - x\right) = 1 - \tan x \) (from the tangent subtraction formula):
\[ I = -\left(1 - \tan x\right) - x + C \]
This simplifies to:
\[ I = -1 + \tan x - x + C \]

### Final Answer
Thus, the final result of the integral is:
\[ I = \tan x - x - 1 + C \]