Question

∫dx/[x(1+x^8)]=_____

Answer 1

1-x^8 = -(1+x^8) +2

∫(1-x^8)/[x(1+x^8)] dx
=-∫(1/x) dx + 2∫dx/[x(1+x^8)]
=-ln|x| +2∫dx/[x(1+x^8)]

let
x^4 = tany
4x^3 dx = (secy)^2 .dy

∫dx/[x(1+x^8)]
=(1/4)∫(4x^3 dx)/[x^4.(1+x^8)]
=(1/4)∫ (secy)^2 .dy/[tany(secy)^2]
=(1/4)∫ dy/tany
=(1/4)ln|siny| + C'
=(1/4)ln| x^4/√(1+x^8)| + C'

∫(1-x^8)/[x(1+x^8)] dx
=-ln|x| +2∫dx/[x(1+x^8)]
=-ln|x| +(1/2)ln| x^4/√(1+x^8)| + C