$\int_0^{1000} e^{x-[x]} \, dx$ is equal to

The value of $\int_2^3 {\frac{{\sqrt x }}{{\sqrt {5 - x} + \sqrt x }}} \,dx$ is

If $I_{n+1} = \int_{0}^{1} \frac{x^{n+1} - 1}{x + 1} dx$,then the value of $I_{10} + I_{11} + 2 \log 2$ is:

Let $I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x}\left(\frac{3}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. Given that $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ and $\tan ^{-1}(\sqrt{3})=\frac{\pi}{3}$. Then $3 I^2=$

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 ..... .

If $\int_{0}^{1} \tan ^{-1} x \, dx = p$,then the value of $\int_{0}^{1} \tan ^{-1}\left(\frac{1-x}{1+x}\right) \, dx$ is