Question Number 101271 by mathmax by abdo last updated on 01/Jul/20

find ∫   ((xdx)/((√(x^2 +x+1))+(√(x^2 −x+1))))

Commented byDwaipayan Shikari last updated on 01/Jul/20

∫((x((√(x^2 +x+1))−(√(x^2 −x+1))))/(2x))  (1/2)∫(√(x^2 +x+1))−(1/2)∫(√(x^2 −x+1))  (1/2)∫(√((x+(1/2))^2 +(((√3)/2))^2 ))dx−(1/2)∫(√((x−(1/2))^2 +(((√3)/2))^2    )){suppose x+(1/2)=t  (1/2)∫(√(t^2 +(((√3)/2))^2 ))dt−(1/2)∫(√(m^2 +(((√3)/2))^2 ))dm       {suppose x−(1/2)=m  (t/4)(√(t^2 +(3/4)))+(3/(16))log(t+(√(t^2 +(3/4))))−(t/2)(√(m^2 +(3/4)))−(3/(16))log(m+(√(m^2 +(3/4))))  ((2x+1)/8)(√(x^2 +x+1))+(3/(16))log(x+(1/2)+(√(x^2 +x+1)))−((2x−1)/8)(√(x^2 −x+1))−(3/(16))log(x−(1/2)+(√(x^2 −x+1)))+C

Commented bymathmax by abdo last updated on 02/Jul/20

thank you sir.