I am trying to solve find a solution to a). i can use maple to solve it, but does anyone know how. The "sum of" part is confusing me. Thanks in advance for any help.
The integral of $\displaystyle \delta \left({f \left({x}\right)}\right)$ is just $\displaystyle f \left({x}\right)$ 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.
I'd suggest using solving using the method of Laplace transforms.
@Matt: I'm not so sure about the result you state. My memory is that $\displaystyle \delta (g(x)) = \sum_{i} \frac{\delta (x - x_i)}{g'(x_i)}$ where the $\displaystyle x_i$ are the real roots of g(x). Therefore the integral is $\displaystyle \sum_{i} \frac{1}{g'(x_i)}$.
You should be familiar with the Dirac delta function and its properties, in particular the sifting property: $\displaystyle \int_{-\infty}^{+\infty} \delta (t - a) \, f(t) \, dt = f(a)$.
You should also be familiar withy the definition of a Laplace transform: $\displaystyle LT[f(t)] = \int_{0}^{+\infty} e^{-st} f(t) \, dt$.
Using the above you should be able to show that $\displaystyle LT[\delta (t - a)] = e^{-as}$.
Alternatively you can always refer to a standard table of Laplace transforms.
Therefore the Laplace transform of your function is $\displaystyle \sum_{k=0}^{K} \left(e^{-vs/d}\right)^k$.
Use the formula for the sum of a geometric series to evaluate this sum.