The integral of is just if I remember it correctly ...
So see which instances of the delta function are included within the limits of the integral and just add up the occurrences of f(x) at those instances.
You should be familiar with the Dirac delta function and its properties, in particular the sifting property: .
You should also be familiar withy the definition of a Laplace transform: .
Using the above you should be able to show that .
Alternatively you can always refer to a standard table of Laplace transforms.
Therefore the Laplace transform of your function is .
Use the formula for the sum of a geometric series to evaluate this sum.