$\int_0^4 \frac{x+2}{\sqrt{4x-x^2}} dx =$

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

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

The value of $\int_0^{\infty} \frac{dx}{(x^2+4)(x^2+9)}$ is

$\int_0^{\frac{\pi}{2}} x^3 \sin x \, dx =$

$\int_{0}^{\frac{\pi}{2}} \sqrt{\cos \theta} \cdot \sin^{3} \theta d \theta = . . . . . .$