We know, $- \sqrt {{a^2} + {b^2}} \le a\cos x + b\sin x \le \sqrt {{a^2} + {b^2}}$ $\therefore$ $- \sqrt {{3^2} + {4^2}} \le 3\cos x + 4\sin x \le \sqrt {{3^2} + {4^2}}$ $- 5 \le k + 1 \le 5$ $- 6 \le k \le 4$ $\therefore$ Set of integers = $- 6, - 5, - 4, - 3, - 2, - 1,0,1,2,3,4$ = Total 11 intergers.