How do I integrate the result of a DSolve operation?

Question

I've got the following result from evaluating a DSolve problem:

input[] = diffEq = Derivative[1][r][t] == (2*(c + A*t)*(1 + z))/(2 + z);
          initialConditions = r[0] == 0; 
          solution = DSolve[{diffEq, initialConditions}, r, t]
output[] = {{r -> Function[{t}, ((2*c*t + A*t^2)*(1 + z))/(2 + z)]}}

How do I use the output of this, which should be a some function r[t] in an integral like this:

input[] = Integrate[r[t], {t, t1, t0}]

I can't seem to tease the output of the DSolve into a function that can be used in the integral.

Answer 1

Try this:

diffEq = Derivative[1][r][t] == (2*(c + A*t)*(1 + z))/(2 + z);
initialConditions = r[0] == 0;
sol = DSolve[{diffEq, initialConditions}, r, t][[1, 1, 2]];
Integrate[sol[t], {t, t1, t0}]

(*  ((3 c (t0^2 - t1^2) + A (t0^3 - t1^3)) (1 + z))/(3 (2 + z))  *)

Have fun!

Answer 2

Expand your ode by R[t]==r'[t]

{ rr, RR} =DSolveValue[{{diffEq, R'[t] == r[t]}, initialConditions}, {r, R}, t]
    (*
    {Function[{t}, ((2 c t + A t^2) (1 + z))/(2 + z)], 
     Function[{t}, (
      3 c t^2 + A t^3 + 3 c t^2 z + A t^3 z + 6 C[2] + 3 z C[2])/(
      3 (2 + z))]}
*) 

RR'[t] == rr[t] // Simplify
(* True*)