Question

$\int_0^{\pi / 4} \log (1+\tan x) \text{d}x=$

• A. $\frac{\pi}{16} \log 2$
• B. $\frac{\pi}{4} \log 2$
• C. $\frac{\pi}{8} \log 2$
• D. $\pi \log 2$

Hint:

This is one of the most famous standard results in calculus. Memorizing $\int_0^{\pi/4} \log(1+\tan x)\text{d}x = \frac{\pi}{8}\log 2$ will save you several minutes on competitive exams!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks us to evaluate a classic definite integral involving a logarithmic trigonometric function from $0$ to $\frac{\pi}{4}$.

Step 2: Key Formula or Approach:
We use King's Property of definite integrals, which states: $$ \int_a^b f(x) \text{d}x = \int_a^b f(a + b - x) \text{d}x $$ For our limits, this substitution becomes $x \rightarrow \frac{\pi}{4} - x$.

Step 3: Detailed Explanation:
Let the given integral be $I$: $$ I = \int_0^{\pi / 4} \log (1+\tan x) \text{d}x \quad \text{--- (Equation 1)} $$ Applying the integral property: $$ I = \int_0^{\pi / 4} \log \left[ 1 + \tan\left(\frac{\pi}{4} - x\right) \right] \text{d}x $$ Using the trigonometric identity $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$: $$ \tan\left(\frac{\pi}{4} - x\right) = \frac{1 - \tan x}{1 + \tan x} $$ Substitute this back into the expression: $$ I = \int_0^{\pi / 4} \log \left[ 1 + \frac{1 - \tan x}{1 + \tan x} \right] \text{d}x $$ Taking the common denominator inside the logarithm: $$ I = \int_0^{\pi / 4} \log \left[ \frac{1 + \tan x + 1 - \tan x}{1 + \tan x} \right] \text{d}x = \int_0^{\pi / 4} \log \left( \frac{2}{1 + \tan x} \right) \text{d}x $$ Using the logarithmic property $\log\left(\frac{a}{b}\right) = \log a - \log b$: $$ I = \int_0^{\pi / 4} \log 2 \ \text{d}x - \int_0^{\pi / 4} \log(1 + \tan x) \text{d}x $$ Notice that the second integral is identical to our original definition of $I$: $$ I = \log 2 \ [x]_0^{\pi/4} - I $$ $$ 2I = \frac{\pi}{4} \log 2 \implies I = \frac{\pi}{8} \log 2 $$

Step 4: Final Answer:
The value of the definite integral is $\frac{\pi}{8} \log 2$, corresponding to option (C).