Using Integrate to define a function

Question

I'd like to define a function by way of the output of a definite integral with symbolic bounds. For instance, F[m_,k_] := Integrate[x,{x,m,k}] would define F[m_,k_] := (1/2)(k^2 - m^2) This seems to work fine for simple examples, but I've run into the cases where the evaluation takes much longer than the indefinite integral. For instance running

Integrate[r/((p - z)^2 + r^2)^(3/2), {z, 0, L}]

takes several minutes to compute and seems to give different output sometimes. However running the indefinite integral

Integrate[r/((p - z)^2 + r^2)^(3/2), z]

returns almost immediately. How can I use an indefinite integral to generate a function for me? I've tried syntax along the form

F[r_, z_] := Integrate[r/((p - z)^2 + r^2)^(3/2), z]
F[r, L] - F[r, 0]`

but that just takes the indefinite integral of a variable called L and runs in to trouble with the number (it outputs some integral with respect to 0).

Answer 1

This works much faster:

f[r_, z_, from_, to_] := (int = Integrate[r/((l - z)^2 + r^2)^(3/2), z]; 
                           Limit[int, z -> to] - Limit[int, z -> from])

sol = f[r, z, 0, L0]

The reason is takes much longer when doing definite integration directly is due to assumptions. If you gives assumptions, then it will be fast also. Try this:

Assuming[Element[{l, r, L0}, Reals] && l > 0 && r > 0 && L0 > 0, 
      Integrate[r/((l - z)^2 + r^2)^(3/2), {z, 0, L0}]]

Now it gives same result, but much faster

Simplify[% - sol]
(* 0 *)