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Question Number 126990 by physicstutes last updated on 26/Dec/20

$$\mathrm{Obtain}\mathrm{a}\mathrm{formula}\mathrm{for}$$$$I_n=\underset{0}{\stackrel{n}{\int }}\left[x\right]dx\mathrm{in}\mathrm{terms}\mathrm{of}n$$$$\mathrm{where}\left[x\right]\mathrm{is}\mathrm{the}\mathrm{greatest}\mathrm{integer}\mathrm{function}\mathrm{of}x.$$

Answered by mr W last updated on 26/Dec/20

$$I_n=\underset{0}{\stackrel{n}{\int }}\left[x\right]dx$$$$=\underset{k=0}{\sum ^{n-1}}{\int }_k^{k+1}\left[x\right]dx$$$$=\underset{k=0}{\sum ^{n-1}}k{\int }_k^{k+1}dx$$$$=\underset{k=0}{\sum ^{n-1}}k=\frac{n(n-1)}{2}$$

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