#I =int e^x/(x(e+1))dx = 1/((e+1))int (de^x)/(x)#
Perform integration by part to get
Let # u=e^x; v = 1/x#
#int (de^x)/x = e^x/x + int(de^x)/x^2#
Repeat the process of integration by parts,
#int (de^x)/x = e^x/x + e^x/x^2 +int 2/x^3 de^x #
#int (de^x)/x = e^x/x + e^x/x^2 +2/x^3e^x + 6/x^4e^x+24/x^5e^x + 120/x^5e^x+...#
#int (de^x)/x = e^x(1/x + 1/x^2 +2/x^3 + 6/x^4+...+((n-1)!)/x^n + ... )=#
#int (de^x)/x = e^x/x(1+ 1/x +2/x^2 + 6/x^3+...+ (n!)/x^n + ... )##=E_i(x)#
Therefore,
#I = e^x/((e+1))(1/x + 1/x^2 +2/x^3 + 6/x^4+...+((n-1)!)/x^n + ... )#
#I = e^x/((e+1)x)(1+ 1/x +2/x^2 + 6/x^3+...+ (n!)/x^n + ... )#
#I = ( E_i(x))/((e+1))#
where Ei is a form of differential integral.
