Number of integers in the range of 'a' so that the equation $x^3 -3x+a=0$ has all its roots real and distinct, is _____.

The given equation is $x^3 -3x+a=0$ or $x^3-3x=-a$. We observe that the number of real roots of the
given equation is same as the number of points of intersection the curves $y = x^3 - 3x$ and $y = -a$.

These two curves intersect at three distinct points if $-2<a <2$. So, the integral values of a in this range are -1, 0 and 1.