awk '	/^[^#]/{ # skip commented lines
	if(nfu<NF){nfu=NF} # try to deal with nonconstant field number
	for(i=1;i<=nfu;i++){
	    y[i]=$i   # x is stored in y[1]
	    }
	if (oldx && nfu>0){
	    dxhalf=0.5*(y[1]-oldy[1]) # perhaps a check that dxhalf>0 should be added...
	    for(i=2;i<=nfu;i++){
		integral[i]+=(y[i]+oldy[i])*dxhalf # trapezoidal integration
#		print "debug", y[2],oldy[2],"aa",integral[2],"bb",integral[1]
		}
	    }
	oldx=$1  # only used for triggering the first step
	for(i=1;i<=nfu;i++){
	    oldy[i]=y[i]
	    }
	printf "%s",y[1];
	for(i=2;i<=nfu;i++){printf "\t%20.17g",integral[i];}
	printf "\n"
	}' $*
#	writes the primitives of functions f_i(x) given a x-sorted data file
#          x=$1 f_1(x)=$2 f_2(x)=$3 ...
#	version 1.3 - July 2025 - Nicola Manini