FLOOR AND INTEGER PART PRODUCT DEFINITE INTEGRAL:
$latex \displaystyle I_4= \int_0^x \lfloor x \rfloor \left\{x\right\} dx $ $latex \displaystyle x^2=(\lfloor x \rfloor + \left\{x\right\})^2 = {\lfloor x \rfloor}^{2} +2 \lfloor x \rfloor \left\{x\right\} + \left\{x\right\}^2 $
$latex \displaystyle \lfloor x \rfloor \left\{x\right\}=\frac{1}{2}(x^2 - \left\{x\right\}^2 - {\lfloor x \rfloor}^{2} )$
$latex \displaystyle I_4=\frac{1}{2} ( \frac{x^3}{3} - I_3 - I_1)$
$latex \displaystyle I_4=\frac{1}{2} ( \lfloor x \rfloor \left\{x\right\}^2 +\frac{{\lfloor x \rfloor}^{2}-\lfloor x \rfloor}{2}) $
The same result can be derived just adding the areas under the curve.
