The value of \(\int_{0}^{\frac{\pi}{4}}\displaystyle\tan x\,dx\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{2}\log_e 2\)
Since \(\int\tan x\,dx = \log_e|\sec x| + C\),
\( \displaystyle \int_{0}^{\pi/4}\tan x\,dx = \left[\log_e\sec x\right]_{0}^{\pi/4} \)
\( = \log_e\sqrt{2} - \log_e 1\)
\( \displaystyle = \frac{1}{2}\log_e 2 \)
Why the other options are incorrect
Option (1) \(\log_e 2\) — Incorrect. It takes \(\sec\frac{\pi}{4}\) as 2; in fact \(\sec\frac{\pi}{4} = \sqrt{2}\).
Option (2) 1 — Incorrect. This is \(\int_{0}^{\pi/4}\sec^{2}x\,dx\), not the integral of \(\tan x\).
Option (3) \(-\displaystyle\frac{1}{2}\log_e 2\) — Incorrect. It takes \(\log_e\cos x\) without the minus sign; \(\tan x \gt 0\) on \((0, \frac{\pi}{4})\), so the value is positive.