Интеграл (dx)/(x^(1/3)*(x^(1/3)-1))

Замена: x^(1/3) = t; x = t^3; dx = 3 *t^2 * dt.

I = ∫(dx) / (x^(1/3) * (x^(1/3) - 1)) = ∫(3 * t^2 * dt) / (t * (t - 1)) = 3 * ∫(t * dt) / (t - 1) = 3 * ∫((t - 1 + 1) * dt) / (t - 1) = 3 * [∫dt + ∫(dt) / (t - 1)] = 3 * [∫dt + ∫(d(t - 1)) / (t - 1)] = 3 * (t + ln|t - 1|) + C.

Обратная замена: t = x^(1/3).

I = 3 * (x^(1/3) + ln|x^(1/3) - 1|) + C.