How to do symbolic definite integral without copy and paste the intermediate results?

Question

I am do a symbolic definite integral, I am doing it like this:

Integrate[1/Sqrt[(ϵ + u)^2 + Δ^2], ϵ] // FullSimplify

It gives an output:

Log[u + ϵ + Sqrt[Δ^2 + (u + ϵ)^2]]

then I copy this output and define a function respect to the integrate variable:

F2[ϵ_] := Log[u + ϵ + Sqrt[Δ^2 + (u + ϵ)^2]]

Finally I can substitute the upper and lower limit to F2 and obtain my results:

F2[ξ - u] - F2[m]

I don't like the manually copy step. So how do you do this automatically?

Answer 1

Maybe you would like to automate the process even more, by defining a function that does all the steps in your calculation at once:

definiteIntegral[integrand_, {x_, xMin_, xMax_}] := 
 Module[{antiDerivative = Integrate[integrand, x]}, 
  Simplify[Subtract @@ (antiDerivative /. x -> {xMax, xMin})]]

definiteIntegral[
 1/Sqrt[(ϵ + u)^2 + Δ^2], {x, m, ξ - u}]

(* ==> (-m - u + ξ)/Sqrt[Δ^2 + (u + ϵ)^2] *)

As the name implies, definiteIntegral does the definite integral of the expression in the first argument, but unlike Mathematica's built-in integration algorithm, it doesn't get slowed to a crawl by attempting to analyze the integration limits (given as the second argument together with the integration variable).

I took this approach from this answer.

Answer 2

You can do this:

In[20]:= Integrate[1/Sqrt[(ϵ + u)^2 + Δ^2], ϵ] // FullSimplify

Out[20]= Log[u + ϵ + Sqrt[Δ^2 + (u + ϵ)^2]]

In[21]:= F2[ϵ_] = %

Out[21]= Log[u + ϵ + Sqrt[Δ^2 + (u + ϵ)^2]]

Note that I used =, not := to make sure % gets expanded before the definition is recorded.