∫x/(x^2＋x＋1) dx
= (1/2)∫ dln(x^2+x+1) -(1/2)∫1/(x^2＋x＋1) dx
= (1/2)ln(x^2+x+1) -(1/2)∫1/(x^2＋x＋1) dx
x^2＋x＋1= (x+1/2)^2+3/4
let
x+1/2 = (√3/2) tana
dx =(√3/2) (seca)^2 da
∫1/(x^2＋x＋1) dx
=∫(1/[(3/4)(seca)^2] ) (√3/2) (seca)^2 da
=(2√3/3)∫ da
=(2√3/3)a + C'
=(2√3/3) arctan[(2x+1)/√3]+ C'
∫x/(x^2＋x＋1) dx
= (1/2)ln(x^2+x+1) -(1/2)∫1/(x^2＋x＋1) dx
= (1/2)ln(x^2+x+1) -(√3/3) arctan[(2x+1)/√3]+ C