$\begin{array}{l}Thenumberof\mathrm{int}egralvaluesofkforwhichtheequation7\cos x+5\sin x=2k+1\\ hasasolutionis\mathrm{...............}\end{array}$

$\begin{array}{l}-\sqrt{7^2+5^2}<7\cos x+5sinx<\sqrt{7^2+5^2}\Rightarrow -\sqrt{74}<2k+1<\sqrt{74}\Rightarrow -8.6<2k+1<8.6\\ \Rightarrow -9.6<2k<7.6\Rightarrow -4.8<k<\mathrm{3.8.}\\ \therefore Theintegralvaluesofkare-4,3,-2,-1,0,1,2,3.\\ \therefore Thenumberof\mathrm{integral}valuesofkis8.\end{array}$