If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral $\int_0^2 {{x^2}[x]\,dx} $ equals (in $/3$)

If $\int_2^e {\left[ {\frac{1}{{\log x}} - \frac{1}{{{{(\log x)}^2}}}} \right]} \,dx = \alpha + \frac{\beta }{{\log 2}},$ then

The integral $\int_0^1 \cot^{-1}(1 + x + x^2) dx$ is equal to:

For a real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$ and $\{x\} = x - [x]$. Let $n$ be a positive integer. Then,$\int_0^n \cos(2 \pi [x] \{x\}) dx$ is equal to

$\int_0^{\pi /8} \cos^3(4\theta) \, d\theta = $

If $I = \int_{0}^{2} e^{x^{4}}(x - \alpha) dx = 0$, then $\alpha$ lies in the interval