Normalized functions


I’m looking for a smart way to define normalized functions. I usually write

f[x_] := f[x] = A Sin[x]/x; 

Then I integrate the function

Integrate[f[x], {x, -\[Infinity], \[Infinity]}, Assumptions -> A > 0], 

take the output and divide f by it

g[x_] :=g[x] = f[x]/(A \[Pi]). 

Is there a better way than this? Thanks