How to remove ConditionalExpression from output

A more general way, which can be used for other functions like Solve, is to use Normal.

Integrate[1/x^s, {x, 1, Infinity}]

(* ConditionalExpression[1/(-1 + s), Re[s] > 1] *)

Normal[%, ConditionalExpression]

(* 1/(-1 + s) *)

Edit

It looks like as of version 10, one can just use the one argument Normal:

Integrate[1/x^s, {x, 1, Infinity}]

(* ConditionalExpression[1/(-1 + s), Re[s] > 1] *)

Normal[%]

(* 1/(-1 + s) *)

You can use GenerateConditions->False to eliminate conditions (assuming they do not affect your planned use):

S[x_, l_] := (C[1] + 
     Integrate[
      E^(2 Sum[t^i/i, {i, 1, l - 1}]) (1 - t)^2*
       Sum[(l - i*2) t^i, {i, 1, l - 1}]/((t - 1) t^l), {t, 1, x}, 
      GenerateConditions -> False]) x^(l - 1)*
   E^(-2 Sum[x^i/i, {i, 1, l - 1}])/(1 - x)^2;
Table[S[x, i], {i, 2, 3}] // TableForm
CoefficientList[Series[S[x, 2], {x, 0, 3}], x]
CoefficientList[Series[S[x, 3], {x, 0, 3}], x]
CoefficientList[Series[S[x, 3][[1]], {x, 0, 3}], x]

It might be late to answer this, but I think it's important to add my approach.

I came to the same problem of having ConditionalExpressions in my outputs.

After googling I came about this solution, by using the Assuming[] function.

An example:

Assuming[Re[s] > 1, Integrate[1/x^s,{x,1,Infinity}]

Which gives:

1/(-1+s)