5. $\left[\frac{\pi}{12}, \frac{\pi}{2}\right]$.
$\frac{1}{4}+x-x^{2}=\frac{1}{2}-\left(x-\frac{1}{2}\right)^{2} \leqslant \frac{1}{2}$, so the range of the original function $y=\frac{1}{2} \arccos \left(\frac{1}{4}+x-x^{2}\right)$ is $\left[\frac{\pi}{12}, \frac{\pi}{2}\right]$.
5. $y=\frac{1}{2} \arccos \left(\frac{1}{4}+x-x^{2}\right)$ The range of values for this function is

保留源文本的换行和格式，直接输出翻译结果如下：

5. $y=\frac{1}{2} \arccos \left(\frac{1}{4}+x-x^{2}\right)$ The range of values for this function is