Let $\alpha > 0$. If $\int \limits _0^\alpha \frac{ x }{\sqrt{ x +\alpha}-\sqrt{ x }} dx =\frac{16+20 \sqrt{2}}{15}$,then $\alpha$ 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.............

$\int_0^{b - c} f''(x + a) \, dx = $

$\int_0^\pi \frac{1}{1+\sin x} dx$ is equal to

$3. \int_0^{\frac{1}{2}} \frac{x \sin^{-1} x}{\sqrt{1-x^2}} dx =$

$\int_2^3 \frac{d x}{x^2-x}$ is equal to