An easier way to do this is to note that, if you have something like Integral{0,x} f(t)dt, this is just the integral of the function f(x). Eg, if f(t) = t^2, this would just be x^3/3. So, if you differentiate it, you'll just get back to the function f(x). Here, Integral{K,Infinity} D0(a)da is the same as -Integral{Infinity,K} D0(a)da, and the letters are different. (We have K instead of x, a instead of t, D0 instead of f and Infinity instead of 0.) But, if you differentiate wrt K, you'll just get -D0(K), as before. This was another very difficult question!
