First slide

Evaluation of definite integrals

Question

If $n \in N$ , the value of $\int_{0}^{n} [x] d x$ (where $(x)$ is the greatest integer function) is

Easy

A

$\frac{n (n + 1)}{2}$
B

$\frac{n (n - 1)}{2}$
C

$n (n - 1)$
D

none of these

Solution

$\begin{array}{l} \int_{0}^{n} [x] d x = \sum_{i = 1}^{n} \int_{i - 1}^{i} [x] d x = \sum_{i = 1}^{n} \int_{i - 1}^{i} (i - 1) d x \\ = \sum_{i = 1}^{n} (i - 1) = \frac{n (n - 1)}{2} . \end{array}$

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