The value of the definite integral $\int_0^1 \frac{x \, dx}{x^3 + 16}$ lies in the interval $[a, b]$. The smallest such interval is

If the value of the integral $\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,dx = \alpha e^{-1} + \beta$,where $\alpha, \beta \in R, 5\alpha + 6\beta = 0$,and $[x]$ denotes the greatest integer less than or equal to $x$; then the value of $(\alpha + \beta)^{2}$ is equal to:

Let $[ t ]$ denote the greatest integer less than or equal to $t$. Then the value of $\int_{1}^{2} |2x - [3x]| dx$ is

Consider the matrices $A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}$,$B = \begin{bmatrix} 20 \\ m \end{bmatrix}$,and $X = \begin{bmatrix} x \\ y \end{bmatrix}$. Let the set of all $m$,for which the system of equations $AX = B$ has a negative solution (i.e.,$x < 0$ and $y < 0$),be the interval $(a, b)$. Then $8 \int_a^b |A| dm$ is equal to.............

If $f(x) = \int_{-1}^{x} |t| dt$,then for any $x \geq 0$,$f(x)$ is equal to

$\int\limits_0^\infty [2e^{-x}]\, dx$,where $[x]$ denotes the greatest integer function,is equal to: