The value of $\int_{0}^{2\pi} [\sin 2x(1 + \cos 3x)] \,dx$,where $[t]$ denotes the greatest integer function,is

If $f:R \to R$ and $g:R \to R$ are one-to-one,real-valued functions,then the value of the integral $\int_{-\pi}^{\pi} [f(x) + f(-x)][g(x) - g(-x)] \, dx$ is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous functions. Then the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} [f(x)+f(-x)][g(x)-g(-x)] \, dx$ is

The value of $\int_{0}^{\pi} x \sin^{3} x \, dx$ is

The value of the integration $\int_{-\pi / 4}^{\pi / 4} (\lambda|\sin x| + \frac{\mu \sin x}{1+\cos x} + \gamma) \, dx$

$\left[ {\sum\limits_{n = 1}^{10} {\int_{ - 2n - 1}^{2n} {{{\sin }^{27}}x\,dx} } } \right] + \left[ {\sum\limits_{n = 1}^{10} {\int_{2n}^{2n + 1} {{{\sin }^{27}}x\,dx} } } \right]$ equals