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Question Number 133426 by metamorfose last updated on 22/Feb/21
∫⌊x⌋dx=?...
$$\int\lfloor{x}\rfloor{dx}=?… \\ $$
Answered by MJS_new last updated on 22/Feb/21
for a<b: ∫_a ^b ⌊x⌋dx=⌊a⌋∫_a ^(⌈a⌉) dx+∫_(⌈a⌉) ^(⌊b⌋−1) ⌊x⌋dx+⌊b⌋∫_(⌊b⌋) ^b dx=  =⌊a⌋(⌈a⌉−a)+((⌊b⌋(⌊b⌋−1))/2)−((⌈a⌉(⌈a⌉−1))/2)+⌊b⌋(b−⌊b⌋)
$$\mathrm{for}\:{a}<{b}:\:\underset{{a}} {\overset{{b}} {\int}}\lfloor{x}\rfloor{dx}=\lfloor{a}\rfloor\underset{{a}} {\overset{\lceil{a}\rceil} {\int}}{dx}+\underset{\lceil{a}\rceil} {\overset{\lfloor{b}\rfloor−\mathrm{1}} {\int}}\lfloor{x}\rfloor{dx}+\lfloor{b}\rfloor\underset{\lfloor{b}\rfloor} {\overset{{b}} {\int}}{dx}= \\ $$$$=\lfloor{a}\rfloor\left(\lceil{a}\rceil−{a}\right)+\frac{\lfloor{b}\rfloor\left(\lfloor{b}\rfloor−\mathrm{1}\right)}{\mathrm{2}}−\frac{\lceil{a}\rceil\left(\lceil{a}\rceil−\mathrm{1}\right)}{\mathrm{2}}+\lfloor{b}\rfloor\left({b}−\lfloor{b}\rfloor\right) \\ $$

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