The number of integral values of x for which the function $\sqrt{\sin \mathrm{x}+\cos \mathrm{x}}+\sqrt{7\mathrm{x}-{\mathrm{x}}^2-6}$ is defined is _______.

$\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\sin \mathrm{x}+\cos \mathrm{x}}+\sqrt{7\mathrm{x}-{\mathrm{x}}^2-6}\\ =\sqrt{\sqrt{2}\sin \left(\mathrm{x}+\frac{\mathrm{\pi }}{4}\right)}+\sqrt{(\mathrm{x}-6)(1-\mathrm{x})}\end{array}$

Now, f(x) is defined if $\sin \left(\mathrm{x}+\frac{\mathrm{\pi }}{4}\right)\ge 0\text{ and }(\mathrm{x}-6)(1-\mathrm{x})\ge 0$

$\text{ i.e., }0\le \mathrm{x}+\frac{\mathrm{\pi }}{4}\le \mathrm{\pi }\text{ or }2\mathrm{\pi }\le \mathrm{x}+\frac{\mathrm{\pi }}{4}\le 3\mathrm{\pi }\text{ and }1\le \mathrm{x}\le 6$

$\text{ i.e., }-\frac{\mathrm{\pi }}{4}\le \mathrm{x}\le \frac{3\mathrm{\pi }}{4}\text{ or }\frac{7\mathrm{\pi }}{4}\le \mathrm{x}\le \frac{11\mathrm{\pi }}{4}\text{ and }1\le \mathrm{x}\le 6$

$\text{ or }\mathrm{x}\in \left[1,\frac{3\mathrm{\pi }}{4}\right]\cup \left[\frac{7\mathrm{\pi }}{4},6\right]$

Integral values of x are x = 1, 2, and 6.