The description of applyop can be found in its help page, ?applyop. Your example is just a double integral. One doesn't need dsolve to evaluate it. Using applyop, it can be done as

u:=1+sum((-1)^(i)*Heaviside(t-T[i+1]),i=1..n-1)+
sum((-1)^(i)*Heaviside(t-T[n+i]),i=1..n):

value(map[3](applyop,student[Doubleint],1,u,t,t))/M;

  /       /n - 1
  |  2    |-----
  | t     | \        i                               2
  |---- + |  )   (-1)  (1/2 Heaviside(t - T[i + 1]) t
  | 2     | /
  |       |-----
  \       \i = 1

                                               2
         - 1/2 Heaviside(t - T[i + 1]) T[i + 1]  - T[i + 1] (

        Heaviside(t - T[i + 1]) t - Heaviside(t - T[i + 1]) T[i + 1])

         \   /  n
         |   |-----
         |   | \        i                               2
        )| + |  )   (-1)  (1/2 Heaviside(t - T[n + i]) t
         |   | /
         |   |-----
         /   \i = 1

                                               2
         - 1/2 Heaviside(t - T[n + i]) T[n + i]  - T[n + i] (

        Heaviside(t - T[n + i]) t - Heaviside(t - T[n + i]) T[n + i])

         \\
         ||
         ||
        )||/M
         ||
         ||
         //

If you want to avoid using applyop or double integral for some reason, then you could just do your manipulations with dsolve for every term (1 and expressions inside sums) and then add them manually.

_________
Alec Mihailovs
http://mihailovs.com/Alec/