Evaluate the definite integral $\int_{0}^{\frac{2}{3}} \frac{d x}{4+9 x^{2}}$.

If $\beta + 2 \int\limits_0^1 {{x^2}\,{e^{ - {x^2}}}}\, dx = \int\limits_0^1 {{e^{ - {x^2}}}}\, dx$,then the value of $\beta$ is

$\int_0^{\pi /2} {\sqrt {\cos \theta } {{\sin }^3}\theta } \,d\theta = $

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

The value of the definite integral $\int_{0}^{\frac{\pi}{2}} \sin x \sin 2x \sin 3x \, dx$ is equal to:

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.............