In[1]:= A = 1.5;
        B = 2;
In[3]:= Comatrix = {
    {-(A + 2*B), 3*B, 0, 0, 0, 0, 0, 0, 0},
    {A, -(A + 3*B), 4*B, 0, 0, 0, 0, 0, 0},
    {0, A, -(A + 4*B), 5*B, 0, 0, 0, 0, 0},
    {0, 0, A, -(A + 5*B), 6*B, 0, 0, 0, 0},
    {0, 0, 0, A, -(A + 6*B), 7*B, 0, 0, 0},
    {0, 0, 0, 0, A, -(A + 7*B), 8*B, 0, 0},
    {0, 0, 0, 0, 0, A, -(A + 8*B), 9*B, 0},
    {0, 0, 0, 0, 0, 0, A, -(A + 9*B), 10*B},
    {0, 0, 0, 0, 0, 0, 0, A, -(10*B)}
       };
In[4]:= Nmatrix[t_] = {N2[t], N3[t], N4[t], N5[t], N6[t], N7[t], N8[t],
    N9[t], N10[t]};

In[5]:= system = Derivative[1][Nmatrix][t] == Comatrix . Nmatrix[t];

In[6]:= sol = DSolve[{system, N2[0] == 1, N3[0] == 0, N4[0] == 0, N5[0]
    == 0, N6[0] == 0, N7[0] == 0, N8[0] == 0, N9[0] == 0, N10[0] == 0},
    {N2, N3, N4, N5, N6, N7, N8, N9, N10}, t];

In[7]:= {N2ans[t_], N3ans[t_], N4ans[t_], N5ans[t_], N6ans[t_],
    N7ans[t_], N8ans[t_], N9ans[t_], N10ans[t_]} = {N2[t], N3[t],
    N4[t], N5[t], N6[t], N7[t], N8[t], N9[t], N10[t]} /. Flatten[sol];

In[8]:= f[t_] = 2*B*N2ans[t];
    Integrate[t*f[t], {t, 0, Infinity}]

Out[9]= 0.326222