Best How To :

1. numerical integration always return just a number

   • if you do not want the number but function instead
   • then you can not use numerical integration for this task directly

2. Polynomial approach

   • you can use any approximation/interpolation technique to obtain a polynomial representing f(x)
   • then integrate as standard polynomial (just change in exponent and multiplication constant)
   • this is not suited for transcendent, periodical or complex shaped functions
   • most common approaches is use of L'Grange or Taylor series
   • for both you need a parser capable of returning value of f(x) for any given x

3. algebraic integration

   • this is not solvable for any f(x) because we do not know how to integrate everything
   • so you would need to program all the rules for integration
   • like per-partes,substitutions,Z or L'Place transforms
   • and write a solver within string/symbol paradigm
   • that is huge amount of work
   • may be there are libs or dlls that can do that
   • from programs like Derive or Matlab ...

[edit1] As the function f(x) is just a table in form

• double f[][2]={ x1,f(x1),x2,f(x2),...xn,f(xn) };
• you can create the same table for g(x)=Integral(f(x)) at interval <0,x>

• so:

  g(x1)=f(x1)*(x1-0)
  g(x2)=f(x1)*(x1-0)+f(x2)*(x2-x1)
  g(x3)=f(x1)*(x1-0)+f(x2)*(x2-x1)+f(x3)*(x3-x2)
  ...
  

• this is just a table so if you want actual function you need to convert this to polynomial via L'Grange or any other interpolation...

• you can also use DFFT and for the function as set of sin-waves