La evaluación de $\displaystyle \int _{-100}^{100}[x^3]\,dx$ donde $[x]$ es la función del suelo

Question

Hay una alternativa mejor solución?

$I=\displaystyle \int _{-100}^{100}[x^3]\,dx$ $=\displaystyle \int _{-100}^{100}[(100-100-x)^3]\,dx$ $\quad$ [$\because \int_{a}^{b}f(x)\,dx=\int_{a}^{b}f(a+b-x)\,dx$]

$=\displaystyle \int _{-100}^{100}[-x^3]\,dx$

$=\displaystyle \int _{-100}^{100}(-[x^3]-1)\,dx$ $\quad$ [$\because [x]+[-x]=-1$ al $x\notin \mathbb{Z}$]

$\Rightarrow I=-I-200$ $\quad$ $\Rightarrow I=-100$

Answer 1

Para $x>0$ hemos $[-x]=-[x]-1.$ $$\text {So }\quad \int_{-100}^0[y]^3\;dy = \int_0^{100}(-[x^3]-1)\;dx$$ is obtained by letting $y=-x.$ $$\text { Therefore } \quad \int_{-100}^{100}[x^3]\;dx=\int_{-100}^0 [y^3]\;dy+\int_0^{100}[x^3]\;dx=$$ $$=\int_0^{100}(-[x^3]-1)\;dx+\int_0^{100} [x^3]\;dx=$$ $$=\int_0^{100} (-1-[x^3]+[x^3])\;dx=\int_0^{100}(-1)\;dx=-100.$$