$\int_{-1}^{1} [x + [x + [x]]] \, dx = $ (where $[\cdot]$ denotes the greatest integer function)

The value of the integral $\int_{1}^{3} [x^{2}-2x-2] dx$,where $[x]$ denotes the greatest integer function,is:

For a real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. The smallest positive integer $n$ for which the integral $\int_{1}^{n} [x][\sqrt{x}] \, dx$ exceeds $60$ is

The positive value of $x$ satisfying the equation $\int_x^1(1-t) dt = \frac{1}{2}$ is

If $\int_a^b x^3 dx = 0$ and $\int_a^b x^2 dx = \frac{2}{3}$,then $a$ and $b$ are respectively

If $\int_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{(1+x^{2})^{3}}}} dx = \alpha \sqrt{2} + \beta \sqrt{3}$,where $\alpha, \beta$ are integers,then $\alpha + \beta$ is equal to.