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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ \begin{align} \int x^{\mu}\,\dd x&={x^{\mu + 1} \over \mu + 1}\ \imp\ \int x^{\mu}\ln\pars{x}\,\dd x=-\,{x^{1 + \mu} \over \pars{1 + \mu}^{2}} + {x^{1 + \mu}\ln\pars{x} \over 1 + \mu} \end{align}

Set $\ds{\mu = -\,{1 \over 3}}$: $$\color{#00f}{\large% \int {\ln\pars{x} \over \root[3]{x}}\,\dd x=-\,{9x^{1/3} \over 4} + {3x^{2/3}\ln\pars{x} \over 2}} + \mbox{a constant} $$