Question Number 148564 by mathmax by abdo last updated on 29/Jul/21

calculate ∫_1 ^2 ((logx)/(1+x))dx

Answered by Kamel last updated on 29/Jul/21

Ω=∫_1 ^2 ((Ln(x))/(1+x))dx=−∫_(1/2) ^1 ((Ln(x))/(x(1+x)))dx =(1/2)Ln^2 (2)+Ln(2)Ln((3/2))−∫_(1/2) ^1 ((Ln(1+x))/x)dx =Ln(2)Ln(3)−(1/2)Ln^2 (2)+Li_2 (−1)−Li_2 (−(1/2)) ∴∫_1 ^2 ((Ln(x))/(1+x))dx =Ln(2)Ln(3)−(1/2)Ln^2 (2)−(π^2 /(12))−Li_2 (−(1/2))