Hey ! what is the integral of x^(1/3)/x^(1/3)-1

substitute x^1/3 = u

\[\int\limits_{}^{}\frac u {u-1}\]

\[\int\limits_{?}^{?} \sqrt[3]{x}\div \sqrt[3]{x}-1\]

yeah

how would u find the deriv of u over u = or - n

use the quotient rule or something. I don't remember, I did this in high school

let me find it and paste it here

alrighty..cool:)

u/(u-1) =1 + 1/(u-1)

ohh...how did u get tht?

there is a technique for it but I dont know that, I just always look at it and think :-)

\[\int\limits \frac{x^{1/3}}{x^{1/3}-1}dx\] Do a u-sub: u=x^(1/3)=>x=u^3 dx=3u^2du \[\int\limits \frac{(u)(3u^2)du}{u-1}=3 \int\limits \frac{u^3}{u-1}du\] From here, you can do long division. Then it should fall out.

Let me know if you want me to take it further.

take it further plz:)

u=x^(1/3) du=1/3 x^(-2/3) dx isnt this the next step?

You can solve for u and find du that way or do it that way. That way is much more complicated^^.

both ways shown wud b gr8!

After doing long division the integral gets simplified to this: \[\int\limits u^2+u+1+\frac{1}{u-1} du\] Integrate term by term giving: \[\frac{u^3}{3}+\frac{u^2}{2}+u+\ln|u-1|+C\] Then plug back in. U=x^(1/3) so this becomes: \[\frac{x}{3}+\frac{1}{2}x^{2/3}+x^{1/3}+\ln|x^{1/3}-1|+C\]

Then of course, all of that its multiplied by 3 (I forgot to copy it).

Wolfram concurs.

ohhhh...got it! thank you so much! i wish my teacher explained them well!:

Youre welcome :D You can post as many as you need help with. I love calc haha

I would probably prefer a u-sub of \[u = x^{1/3} - 1\]\[ \implies du = {1 \over 3x^{2/3}}dx\]\[\implies dx = 3(u+1)^2du\]\[\implies \int {x^{1/3}\over x^{1/3} - 1}dx = 3\int {(u+1) \over u} (u+1)^2 du = 3\int {(u+1)^3 \over u} du\] ... etc. Though I wasn't really that strong at polynomial division when I was doing this stuff and I'd probably try your way instead.

Now that I'm much better at it I mean.

I know polpak :DDD haha

Thats the nice thing about integrals, they take some thought sometimes xD

haha i hate giving so much thought in math..plug in formulas rock!:) im prob ganna need help with tons more..got my finals on this stuff n derivatives n limits tom!

Just keep posting in the feed, I'll be on for a while xP

Careful though.. the first one is always free.. after that we make you work for it ;p

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