Q:

Evaluating integral with $\sqrt[3] x$ and $\ln x$

I found this example amongst set of examples to get ready on exam
$$
\int\frac{\ln x}{\large\sqrt[3]{x}}\ dx
$$
I am able to see the basic substitution, but I don't know how to count if further… Anyone who could help me with this one?
Thank you.

A:

Hint
Use integration by parts with $u=\log(x)$ and $v'=x^{-1/3}~dx$. So, $u'=\frac {dx}{x}$ and $v=\frac{3 x^{2/3}}{2}$.
I am sure that you can take from here.

A:

Using IBP, let $u=\ln x$, $du=\dfrac1x\ dx$, $dv=x^{-\Large\frac13}\ dx$, and
$$
v=\int x^{-\Large\frac13}\ dx=\frac32x^{\Large\frac23},
$$
then
$$
\int\frac{\ln x}{\sqrt[3]{x}}\ dx=\frac32x^{\Large\frac23}\ln x-\int\frac32x^{\Large\frac23}\dfrac1x\ dx=\frac32x^{\Large\frac23}\ln x-\frac32\int x^{-\Large\frac13}\ dx.
$$
I leave it the rest for you.

