First slide

Evaluation of definite integrals

Question

The value of $\sum_{n = 1}^{1000} \int_{n - 1}^{n} e^{x - [x]} d x$ is ([x] is greatest integer function)

Moderate

A

$\frac{e^{1000} - 1}{1000}$
B

$\frac{e^{1000} - 1}{e - 1}$
C

$1000 (e - 1)$
D

$\frac{e - 1}{1000}$

Solution

Since the period of the function $x - [x]$ is 1 so
$\begin{array}{l} \sum_{n = 1}^{1000} \int_{n - 1}^{n} e^{x - [x]} d x = \int_{0}^{1000} e^{x - [x]} d x \\ = 1000 \int_{0}^{1} e^{x - [x]} d x = 1000 \int_{0}^{1} e^{x} d x = 1000 (e - 1) . \end{array}$

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