$\int_0^a f(x) \, dx = $

The value of the integral $\sum\limits_{k = 1}^n {\int_0^1 {f(k - 1 + x)\,dx} } $ is

Let $I_{n} = \int_{1}^{e} x^{19}(\log |x|)^{n} dx$,where $n \in N$. If $(20) I_{10} = \alpha I_{9} + \beta I_{8}$,for natural numbers $\alpha$ and $\beta$,then $\alpha - \beta$ is equal to ..... .

Let $f(x) = \int_{0}^{x} g(t) dt$,where $g$ is a non-zero even function. If $f(x+5) = g(x)$,then $\int_{0}^{x} f(t) dt$ equals

$\int_0^{\frac{\pi}{2}} \frac{300 \sin x+100 \cos x}{\sin x+\cos x} \,dx = \ldots$ (in $\pi$)

$\int_{-1}^1 x|x| \, dx =$