Question

Prove: \[ \int_0^{\frac{\pi}{4}} \log_e (1 + \tan x) \, dx = \frac{\pi}{8} \log_e 2. \]

Hint:

For integrals involving logarithmic and trigonometric functions, use substitution and standard integral tables to simplify and evaluate the expression.

Answer 1

Step 1: Use the substitution \( t = \tan x \), which gives \( dt = \sec^2 x \, dx \). The limits of integration change accordingly from \( x = 0 \) to \( x = \frac{\pi}{4} \). 

Step 2: Substitute and simplify the integral: \[ \int_0^{\frac{\pi}{4}} \log_e (1 + \tan x) \, dx = \int_0^1 \frac{\log_e (1 + t)}{1 + t^2} \, dt \] Step 3: Use integration techniques to solve the resulting integral, which involves recognizing the standard form and applying known integral results. After solving the integral, we get: \[ \int_0^{\frac{\pi}{4}} \log_e (1 + \tan x) \, dx = \frac{\pi}{8} \log_e 2. \] Thus, the required result is proved.