Question

If \(I_n = \int _1^e(log_ex)^n\,dx\) where \(n\in Z^+\) and \(I_m+mI_{2026} = e\), then \(m =\)

• A. \(2024\)
• B. \(2025\)
• C. \(2026\)
• D. \(2027\)

Hint:

Integrate by parts to get \(I_n=e-nI_{n-1}\).

Answer 1

The Correct Option is D

Step 1: Check the recurrence on small n
$I_0=e-1$, $I_1=e-I_0=1$, and by the formula $I_1+1\cdot I_0=1+e-1=e$, consistent.

Step 2: Match
The relation $I_k+kI_{k-1}=e$ holds for every $k$. Taking $k-1=2026$ gives $k=2027$, so $m=2027$, option (D).

Final Answer:
Option (D), $m=2027$. \[ \boxed{2027} \]