First slide

Evaluation of definite integrals

Question

The value of $\sum_{n = 1}^{100} \int_{n - 1}^{n} e^{x - [x]} d x$ , where $[x]$ is the greatest integer less than or equal to x, is :

Moderate

A

$100 (1 - e)$
B

$100 (1 + e)$
C

$100 e$
D

$100 (e - 1)$

Solution

$\sum_{n = 1}^{100} \int_{n - 1}^{n} e^{x - [x]} d x = \sum_{n = 1}^{100} \int_{n - 1}^{n} e^{\{x\}} d x = \sum_{n = 1}^{100} \int_{0}^{1} e^{\{x\}} d x$ ${f o r m u l a \int_{m T}^{n T} f (x) d x = (n - m) \int_{0}^{T} f (x) d x w h e r e T = p e r i o d o f f (x)}$
=    $\sum_{n = 1}^{100} \int_{0}^{1} e^{x} d x$ ${\{x\} i s p e r i o d i c f u n c t i o n w i t h p e r i o d 1}$
= $\sum_{n = 1}^{100} (e - 1)$
   $= 100 (e - 1)$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App