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