四、计算题 (本题满分 8 分)
计算定积分 $\int_{0}^{1} x \arcsin x \mathrm{~d} x$.
$$
I=\int_{0}^{1} x \arcsin x \mathrm{~ d} x=\frac{1}{2} \int_{0}^{1} \arcsin x \mathrm{~ d} \left(x^{2}\right)=
$$
$$
\frac{1}{2}\left[\left.x^{2} \arcsin x\right|_{0} ^{1}-\int_{0}^{1} x^{2} \frac{1}{\sqrt{1-x^{2}}} \mathrm{~ d} x\right]
$$
其中:
$$
\int_{0}^{1} x^{2} \frac{1}{\sqrt{1-x^{2}}} \mathrm{~ d} x=x=\sin t \Rightarrow
$$
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} t}{\cos t} \cdot \cos t \mathrm{~ d} t=\frac{1}{2} \cdot \frac{\pi}{2}=\frac{\pi}{4}
$$
于是:
$$
I=\frac{1}{2}\left[\frac{\pi}{2}-\frac{\pi}{4}\right]=\frac{\pi}{8}
$$