The value of the integral $\int_{-\pi/4}^{\pi/4} \sin^4 x \, dx$ is

$\int_0^{\pi /4} (\cos x - \sin x) dx + \int_{\pi /4}^{5\pi /4} (\sin x - \cos x) dx + \int_{2\pi }^{\pi /4} (\cos x - \sin x) dx$ is equal to

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

Let $f(x) = \max \{x + |x|, x - [x]\}$, where $[x]$ stands for the greatest integer not greater than $x$. Then $\int_{-3}^3 f(x) \, dx$ has the value:

Let the function $f :[0,2] \rightarrow R$ be defined as $f(x)=\begin{cases} e^{\min \{x^2, x-[x]\}}, & x \in[0,1) \\ e^{[x-\log_e x]}, & x \in[1,2] \end{cases}$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int_0^2 x f(x) dx$ is

The value of $\int_{0}^{8} |x - 5| \, dx$ is