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AI-তৈরি নমুনা উত্তর

ধরি,
$I = \int_0^{\frac{\pi}{4}} \ln(1 + \tan x) dx \quad \text{--- (1)}$

নির্দিষ্ট যোগজের ধর্ম অনুসারে, $\int_0^a f(x) dx = \int_0^a f(a - x) dx$
সুতরাং,
$I = \int_0^{\frac{\pi}{4}} \ln\left(1 + \tan\left(\frac{\pi}{4} - x\right)\right) dx$
$= \int_0^{\frac{\pi}{4}} \ln\left(1 + \frac{1 - \tan x}{1 + \tan x}\right) dx$
$= \int_0^{\frac{\pi}{4}} \ln\left(\frac{1 + \tan x + 1 - \tan x}{1 + \tan x}\right) dx$
$= \int_0^{\frac{\pi}{4}} \ln\left(\frac{2}{1 + \tan x}\right) dx$
$= \int_0^{\frac{\pi}{4}} [\ln 2 - \ln(1 + \tan x)] dx$
$= \int_0^{\frac{\pi}{4}} \ln 2 \, dx - \int_0^{\frac{\pi}{4}} \ln(1 + \tan x) dx$
$= \ln 2 [x]_0^{\frac{\pi}{4}} - I$

বা, $2I = \frac{\pi}{4} \ln 2$
$\therefore I = \frac{\pi}{8} \ln 2$

অতএব, $\int_0^{\frac{\pi}{4}} \ln(1 + \tan x) dx = \frac{\pi}{8} \ln 2$ (দেখানো হলো)।