If n ∈ N , the value of ∫0n [x]dx (where (x) is the greatest integer function) is

∫0n [x]dx=∑i=1n ∫i−1i [x]dx=∑i=1n ∫i−1i (i−1)dx=∑i=1n (i−1)=n(n−1)2.

If n ∈ N , the value of ∫0n [x]dx (where (x) is the greatest integer function) is

∫0n [x]dx=∑i=1n ∫i−1i [x]dx=∑i=1n ∫i−1i (i−1)dx=∑i=1n (i−1)=n(n−1)2.

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

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

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

none of these

$n (n - 1)$

answer is B.

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Detailed 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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If n ∈ N , the value of ∫0n [x]dx (where (x) is the greatest integer function) is

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