This problem can be solved using integrate by part.

$$\begin{align*} &\int_0^1 x\arcsin{x} dx \\ &= \int_0^1 \arcsin{x}d\left(\frac{x^2}{2}\right)\\&= \frac{x^2}{2}\arcsin{x}|_0^1 - \int_0^1 \frac{x^2}{2}d(\arcsin{x}) \\& = \frac{\pi}{4} - \frac{1}{2}\int_0^1 \frac{x^2}{\sqrt{1-x^2}} dx\\&= \frac{\pi}{4} +\frac{1}{2}\left(\int_0^1\frac{(1-x^2)-1}{\sqrt{1-x^2}}dx\right) \\&=\frac{\pi}{4} +\frac{1}{2}\left(\int_0^1\sqrt{1-x^2}dx - \int_0^1 \frac{1}{\sqrt{1-x^2}}dx\right) \\&= \frac{\pi}{4} + \frac{1}{2}\left(\frac{\pi}{4} - \arcsin{x}|_0^1 \right)\\&=\boxed{\frac{\pi}{8}} \end{align*}$$