∫ (1 - tan x) / (1 + tan x) d x =

• sec² (π / 4 - x) + c
• log (sec x - sin x) + c
• log (sin x + cos x) - k
• log [sec (π / 4 - x)] + c

To solve the integral ∫ (1 - tan x) / (1 + tan x) dx, we can start by simplifying the integrand. The expression can be rewritten using trigonometric identities.

Step-by-Step Breakdown

First, we can express the tangent function in terms of sine and cosine:

• tan x = sin x / cos x

This allows us to rewrite the integral as:

∫ (cos x - sin x) / (cos x + sin x) dx

Using Substitution

Next, we can use a substitution to simplify the integral further. Let:

• u = cos x + sin x
• du = (cos x - sin x) dx

Now, the integral becomes:

∫ du / u

Integrating

The integral of 1/u is:

ln |u| + C = ln |cos x + sin x| + C

Final Expression

Thus, the integral ∫ (1 - tan x) / (1 + tan x) dx evaluates to:

ln |cos x + sin x| + C

For the specific expression you provided, it seems to relate to a more complex transformation involving secant and logarithmic functions. The constants and additional terms may arise from specific boundary conditions or transformations applied to the integral. Always ensure to check the context of the problem for any additional constraints or requirements.