$\int_0^1 |5x - 3| \, dx = $

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x) = \begin{cases} x^2+3x+a, & x \leq 1 \\ bx+2, & x > 1 \end{cases}$ is differentiable on $\mathbb{R}$. Then,the value of $\int_{-2}^2 f(x) dx$ equals

$\int_0^a x^2 (a^2 - x^2)^{3/2} dx = $

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 :

Let $[.]$ denote the greatest integer function. If $\int_0^{e^3}\left[\frac{1}{e^{x-1}}\right] d x=\alpha-\log _e 2$,then $\alpha^3$ is equal to . . . . . . .

$\int_0^2 |1-x^2| dx = $