Reducing a 4D function into 1D function

[maoxin]

ROOT 5.34
gcc version 4.9.2 (Debian 4.9.2-10)

Hello, everyone! I got a problem. I wrote a function with four variables. And I have their integration ranges in the form of two arrays xmin[4] and xmax [4]. I know we can get the finite integration of this function by ROOT::Math::IntegratorMultiDim just like the below codes. However, I want to know is there any way I can eliminate the first three variable by integration and leave the fourth variable unchanged. Thus changing this four-dimention function into one-dimention function and drawing this as a TF1. My simple codes are as belows:

#include "TMath.h"
#include "Math/WrappedFunction.h"
#include "Math/IntegratorMultiDim.h"
#include <iostream>

Double_t func(const Double_t *x){ 
    return x[0]*x[1]*TMath::Exp(x[2]*x[3]);
}

void test5()
{
    // numeric integration
    //ROOT::Math::WrappedMultiFunction<> f1(func,3);
	ROOT::Math::Functor f1(&func,4);
    ROOT::Math::IntegratorMultiDim ig(f1);
	
    Double_t xmin[4] = {0.0,0.0,0.0,0.0};
    Double_t xmax[4] = {2.0,3.0,4.0,4.0};
	
    Double_t result = ig.Integral(xmin, xmax);
	cout<<result<<endl;
}

This code is for definite integration. And I thought about one way to get it’s dimentions reduced like the following code:

#include "TMath.h"
#include "Math/WrappedFunction.h"
#include "Math/IntegratorMultiDim.h"
#include <iostream>

Double_t func(Double_t *x,Double_t *b){ 
    return x[0]*x[1]*TMath::Exp(x[2]*b[0]);
}

void test5()
{
	int nbins=500;
	TF3* f3=new TF3("f3",func,0.,2.,0.,3.,0.,4.,1);
	TH1F* h=new TH1F("h","",nbins,0.,5.);
	for(int i=0;i<nbins;i++){
		double E_tem=h->GetBinCenter(i+1);
		f3->SetParameter(0,E_tem);
		double integral;
		integral=f3->Integral(0.,2.,0.,3.,0.,4.);
		h->SetBinContent(i+1,integral);
	}
	h->Draw();	
}

However, it will need lots of calculation for my real work and cause new problem. So I am wondering is there a simple way in root to get the dimentions reduced. Thanks very much for your attention!

[Wile_E_Coyote]

Your function seems to be simple enough that you can write an analytical expression for the integral.

[maoxin]

Thanks for your attention ! And actually the function in my real work is more complecated and it needs to be looped for many times. So I want to find a simple way like TF3::Integral() in root . I think there should be something like infinite integration in root, but I can’t find that.
Also, do you have some examples of the analytical expression for integral?

[Wile_E_Coyote]

Lists of integrals
Table of Integrals