Let [•] denote the greatest integer function. If int 0 e 3 [frac 1 e x-1 ] d x=α-log e 2 , then α 3 is equal to \ \ \ \ -.

Mathematics · Definite Integration

JEE Main 2025 — 2 April, Morning Shift — Question 44

Let [•] denote the greatest integer function. If $\int_{0}^{e^{3}}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _{e} 2$ , then $\alpha^{3}$ is equal to $\_\_\_\_$ -.

Answer: 8

Numerical answer — enter this value.

Step-by-step solution

$I=\int_{0}^{e^{3}}\left[\frac{1}{e^{x-1}}\right] d x=\int_{0}^{1-\ln 2} 2 d x+\int_{1-\ln 2}^{1} d x+\int_{1}^{e^{3}} 0 d x$

$=2(1-\ln 2)+(1-(1-\ln 2))+0=2-\ln 2$

$\Rightarrow \alpha=2 \Rightarrow \alpha^{3}=8$

Answer key and solution verified before publishing.

Practise Definite Integration

Start with this question, then two more from the same chapter — with a tutor that explains every step. Free.

Solve a similar one free

Exam: JEE Main 2025
Paper: 2 April, Morning Shift
Subject: Mathematics
Chapter: Definite Integration
Topic: Evaluation of Definite Integrals