Question

求积分 ∫(arctanx)/(x^2(x^2+1))dx

Answer 1

∫(arctanx)/(x^2(x^2+1))dx

let

x=tana

dx = (seca)^2da

∫(arctanx)/(x^2(x^2+1))dx

= ∫ [a/(tana)^2] da

=-∫ ad(cota+a)

= -a(cota+a) + ∫ (cota+a)da

= -a(cota+a) + ln|sina| + a^2/2 + C

=-arctanx( 1/x + arctanx) + ln|x/√(1+x^2) | + (arctanx)^2/2 + C

=-(1/x)arctanx -(arctanx)^2/2 +ln|x/√(1+x^2) |+ C