\(\int\log_e x\,dx\) is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(x(\log_e x)-x+C\)
Write \(\log_e x = (\log_e x)\cdot 1\) and use integration by parts with \(\log_e x\) as the first function:
\( \displaystyle \int\log_e x\cdot 1\,dx = (\log_e x)\cdot x-\int\frac{1}{x}\cdot x\,dx \)
\( \displaystyle = x\log_e x-\int 1\,dx \)
\(= x\log_e x-x+C\)
NCERT (2026-27 edition):
Integration by Parts
