Question

integral of ln times cube root x dx

Answer 1

If u = ln(x^(1/3))
then by the chain rule
du = [1/x^(1/3)] * (1/3)x^(-2/3) dx = (1/3)(1/x) dx. 
You could get there faster by noting that
 ln(x^(1/3)) = (1/3) ln(x), so du = (1/3) dx/x