Integral of ln times cube root x dx

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