You can define a function:

Clear[a, x];
a[x_] = Integrate[x, x]
a[1]

Two responses show how one might do this. Here is a third, and it shows what went awry in the original attempt.

First note that With holds all its arguments.

In[298]:= Attributes[With]

(* Out[298]= {HoldAll, Protected} *)

The practical consequence is that there is no x in the second argument until after it is evaluated. And this happens (as it happens) after the macro-style substitution of 10 for x in that second argument. To force the expression in the second argument to be x^2/2, simply use Evaluate.

In[299]:= With[{x = 10}, Evaluate@a]

Out[299]= 50

This next shows a bit more detail about the evaluation sequence for the example.

In[302]:= a = Integrate[x, x];
Trace[With[{x = 10}, a]]
Trace[With[{x = 10}, Evaluate@a]]

(* Out[303]= {HoldForm[With[{x = 10}, a]], HoldForm[a], HoldForm[x^2/2]}

Out[304]= {{HoldForm[a], HoldForm[x^2/2]}, 
   HoldForm[With[{x = 10}, x^2/2]], HoldForm[10^2/2], 
   {HoldForm[10^2], HoldForm[100]}, HoldForm[100/2], HoldForm[50]} *)

I will remark that similar issues arise with Plot and related functions that hold their arguments.

Use Block instead of With

In[1]:= a = Integrate[x, x];

In[2]:= Block[{x = 10}, a]
Out[2]= 50