The number of integral values of x for which the function sin⁡x+cos⁡x+7x−x2−6 is defined is _______.

f(x)=sin⁡x+cos⁡x+7x−x2−6=2sin⁡x+π4+(x−6)(1−x)Now, f(x) is defined if sin⁡x+π4≥0 and (x−6)(1−x)≥0 i.e., 0≤x+π4≤π or 2π≤x+π4≤3π and 1≤x≤6 i.e., −π4≤x≤3π4 or 7π4≤x≤11π4 and 1≤x≤6 or x∈1,3π4∪7π4,6Integral values of x are x = 1, 2, and 6.