SQUARE FLOOR FUNCTION DEFINITE INTEGRAL:
$latex \displaystyle I_3= \int_0^x \lfloor x \rfloor^2 dx = \int_0^{\lfloor x \rfloor} \lfloor x \rfloor^2 dx+ \int_{{\lfloor x \rfloor}}^ {x} \lfloor x \rfloor^2 dx $
$latex \displaystyle I_3=\sum_{k=1}^{ \lfloor x \rfloor-1}{k^2} + \left\{{x}\right\}\lfloor x \rfloor^2 $
$latex \displaystyle I_3=P(\lfloor x \rfloor-1) + \left\{{x}\right\}\lfloor x \rfloor^2 $
Where: $latex \displaystyle P(n)$ gives the n-th Square Pyramidal Number.
$latex \displaystyle P(n) =\frac{n(n+1)(2n+1)}{6} $
POWER FLOOR FUNCTION DEFINITE INTEGRAL:
$latex \displaystyle I_4= \int_0^x \lfloor x \rfloor^n dx = \sum_{k=1}^{ \lfloor x \rfloor-1}{k^n} + \left\{{x}\right\}\lfloor x \rfloor^n$
$latex \displaystyle S(n,m)=\sum_{k=1}^{m}{k^n} \;\;\;\;$ is the Faulhaber's formula.
If $latex \displaystyle n=1$ , the formula gives the Triangular Numbers.
And if $latex \displaystyle n=2$ , the formula gives the Square Pyramidal Numbers.
