Question Number 104388 by Ar Brandon last updated on 21/Jul/20

∫(dx/(x^2 ((x^3 −1))^(1/3) ))

Answered by Dwaipayan Shikari last updated on 21/Jul/20

∫(dx/(x^3 (1−(1/x^3 ))^(1/3) ))=∫(((1/x^4 ).x)/((1−(1/x^3 ))^(1/3) ))dx=(x/3)∫((3/x^4 )/((1−(1/x^3 ))^(1/3) ))−(1/3)∫∫((3/x^4 )/((1−(1/x^3 ))^(1/3) ))  =(x/2)(1−(1/x^3 ))^(2/3) −(1/2)∫(1−(1/x^3 ))^(2/3)   =(x/2)(1−(1/x^3 ))^(2/3) −(1/2) ∫(3/x^4 ).(x^4 /3)(1−(1/x^3 ))^(2/3) dx  (x/2)(1−(1/x^3 ))^(2/3) −(1/6)∫x^4 t^(2/3) dt       {1−(1/x^3 )=t  x^3 =(1/(1−t))  (x/2)(1−(1/x^3 ))−(1/6)∫((1/(1−t)))^(4/3) t^(2/3)    .....continue