What is the value of the integral \( \int_{0}^{1} \frac{(x-x^3)^{\frac{1}{3}}}{x^4} \, dx \)?

Rewriting numerator as \( x^{1/3}(1-x^2)^{1/3} \), division by \( x^4 \) gives \( \int x^{-8/3}(1-x^2)^{1/3} \, dx \). Substituting \( t = \frac{1}{x^2}-1 \) evaluates to 3/2 or 3 appropriately.