If $\sin (\alpha+\beta)=1, \sin (\alpha-\beta)=\frac{1}{2}, \alpha, \beta \in\left[0, \frac{\pi}{2}\right]$, then $\tan (\alpha+2 \beta) \cdot \tan (2 \alpha+\beta)=$

The value of $\int_0^\pi\left|\sin ^3 x\right| \mathrm{d} x$ is

The population $p$ of the city at time $t$ is given by $\frac{\mathrm{dp}}{\mathrm{dt}}=\frac{\mathrm{p}}{2}-100$. If initial population is 100 then $\mathrm{p}=$

$$ \int_0^1 \log (x+1) d x= $$