This problem appears in another thread and similar ones pop up on contest sites where they are considered tough. One of the oddities that I have noticed is that when a problem is thorny using classical methods it will often succumb to EM rather easily. This is certainly the case here.
Here is the problem:
and we are asked to solve for a,b,c,d,f.
Whatever those coefficients are they might just work for definite integrals too. This reduces the problem to solving a 6 x 6 simultaneous set of linear equations and to the numerical evaluation of 6 integrals. This of course is generally a trivial matter to almost all the software out there.
We obtain the six integrals using Gaussian or Romberg integration.
If we call the RHS of 1) g(x) we have:
We know from calculus that
We can now easily follow the pattern and generate the 6 simultaneous equations.
Solving for a,b,c,d,f we get:
a = -4
b = -12
c = -20
d = 0
f = 32
More to come...