# How to integrate by parts and $u$-sub $8 \ln(\sqrt[3]{x})$?

I am having trouble integrating the following equation by first using u-sub and then integration by parts:

$$\int 8 \ln{\sqrt[3]{x}}~dx$$

I looked up the answer on Wolfram Alpha but it is still unclear to me.

• Hi. Can you show a bit of your work on this so far? This way it is easier to help you! – MattAllegro May 10 '14 at 7:35

Hint

Remember that $\log(x^a)=a \log(x)$. So $$\int 8 \ln{\sqrt[3]{x}}~dx=\frac{8}{3} \int \log(x)~dx$$ Now, use integration by parts.

I am sure that you can take from here.

• You saved my life!!!!!!!!!!!!!!!!!!!! Thank you! Thank you both! – user4681 May 10 '14 at 7:04
• I did not save your life and you are very welcome ! – Claude Leibovici May 10 '14 at 7:06

Call it $\frac{8}{3}\ln x$, and let $u=\ln x$, $dv=\frac{8}{3}\,dx$.

But you can actually integrate directly by parts, $u=8\ln(x^{1/3})$, $dv=dx$.

• Thanks a lot!!!!! It helped me a lot! – user4681 May 10 '14 at 7:06
• You are welcome. In this case, we don't need to bother with substitution, though it is a natural and often useful idea. – André Nicolas May 10 '14 at 7:08