The value of \(\int_{1}^{e}\displaystyle\log_e x\,dx\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1
By integration by parts, taking \(\log_e x\) as the first function and 1 as the second,
\( \displaystyle \int\log_e x\,dx = x\log_e x - \int x\cdot\frac{1}{x}\,dx \)
\( = x\log_e x - x\)
Putting in the limits, with \(\log_e e = 1\) and \(\log_e 1 = 0\):
\( \left[x\log_e x - x\right]_{1}^{e} = (e - e) - (0 - 1)\)
\( = 0 + 1 = 1\)
Why the other options are incorrect
Option (2) \(e-1\) — Incorrect. This is only \(\int_{1}^{e}1\,dx\), the part that is subtracted.
Option (3) e — Incorrect. This is only \(\left[x\log_e x\right]_{1}^{e}\); the term \(\int_{1}^{e}1\,dx = e-1\) must be subtracted.
Option (4) 0 — Incorrect. \(\log_e x \gt 0\) for \(1 \lt x \lt e\), so the integral is positive.