The value of ∑ n = 1 100 ∫ n - 1 n e x - x d x , where x is the greatest integer ≤ x , is:

Open MARKS Web Download App

Search any question & find its solution

Question: Answered & Verified by Expert

The value of

\sum_{n = 1}^{100} \int_{n - 1}^{n} e^{x - [x]} d x

, where

[x]

is the greatest integer

\leq x,

is:

MathematicsDefinite IntegrationJEE MainJEE Main 2021 (26 Feb Shift 1)

Options:

A
$100 (e - 1)$
B
$100 e$
C
$100 (1 - e)$
D $100 (1 + e)$

Solution:

2888 Upvotes Verified Answer

The correct answer is:

100 (e - 1)

\sum_{n = 1}^{100} \int_{n - 1}^{n} e^{\{x\}} d x,

period of

\{x\} = 1

$\sum_{n = 1}^{100} \int_{0}^{1} e^{\{x\}} d x = \sum_{n = 1}^{100} \int_{0}^{1} e^{x} d x$

$\sum_{n = 1}^{100} (e - 1) = 100 (e - 1)$

Looking for more such questions to practice?

Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.

Use on iOS or Desktop Download from Google Play