求定积分区间-2到2∫[(x+|x|)/(2+x^2)]dx,

Question

求定积分 区间-2到2 ∫[(x+|x|)/(2+x^2)]dx,

Answer 1

当-2＜x＜0时
∫[(x+|x|)/(2+x^2)]dx=∫[(x-x)/(2+x^2)]dx=0
当0≤x＜2时
∫[(x+|x|)/(2+x^2)]dx=∫[(x+x)/(2+x^2)]dx=∫[2x/(2+x^2)]dx=∫[1/(2+x^2)]d(2+x^2)=ln(2+x^2)=ln6-ln2=ln3
所以在-2到2区间上
∫[(x+|x|)/(2+x^2)]dx=0+ln3=ln3