Why in the world would you use a "Laplace Transform"?
The definition of the Dirac delta function is that it is the functional that assigns to every function f(x) the value if 0 is in the interval [a, b], 0 otherwise. A simple change of variable extends that to if q is in the interval [a, b], 0 otherwise.
The first integral involves and -1 is in the interval . The integral is .
The second integral involves but 2 is NOT in the interval . The integral is 0.
I have no idea where your "p" came from.