Question

интеграл dx/(x*(lnx+1)^1/3)

Answer 1

$int frac{dx}{xcdot (lnx+1)^{frac{1}{3}}}=int frac{frac{dx}{x}}{(lnx+1)^{frac{1}{3}}}=[, t=lnx+1,; dt=frac{dx}{x}, ]=\\\=int frac{dt}{t^{frac{1}{3}}}=int t^{-frac{1}{3}}, dt=frac{t^{frac{2}{3}}}{frac{2}{3}}+C=frac{3}{2}cdot (lnx+1)^{frac{2}{3}}+C=\\=frac{3}{2}cdot sqrt[3]{(lnx+1)^2}+C$