Question

Int(x^3/(3x-6)) dx
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Answer 1

u=3x-6
du=dx
(u+6)^3/27=x^3 u2+12u+36(u+6)
int1/27 u^3+18u^2+108u+216/u

1/27(u^2+18u+108+216/u)du

then do each individually
1/27(1/3u^3+9u^2+108u+216ln(u)+c

Answer 2

times 1/3 from the du=1/3dx

Answer 3

1/81(1/3u^3+9u^2+108u+216ln(u))+C

Answer 4

i could be very very wrong i didnt do any of this on paper

Answer 5

I would have started with long division to get

\[\frac{x^3}{3x-6}=\frac{8}{3(x-2)}+\frac{x^2}{3}+\frac{2x}{3}+\frac{4}{3}\]

then integrated term by term.

Answer 6

dont really understand whats going on with your u substitution bnut and i have a hard time understanding what you did there Zarkon

Answer 7

ok i get now on paper 1/3( u^3/3+3u^2+12u+8ln(u))

u=x-2
x^3=(x+2)^3

Answer 8

have you done polynomial division before?

Answer 9

yes i have i just now figured out how you did it XD

Answer 10

ok so i said u=3x-6 do du/3=dx

then i isolated x=(u+6)/3 then i cubed that to get x^3

Answer 11

ahh ok i see i see, gotcha

Answer 12

you should end up with


\[\frac{8}{3}\ln(x-2)+\frac{x^3}{9}+\frac{x^2}{3}+\frac{4}{3}x+c\]

Answer 13

ahh i see thats one good way to do it zarkon, wana see bnuts sub way too once hes done with it

Answer 14

ya...it is good to know how to do it both ways.

Answer 15

do you wanna see a good example of it i did it on paper but zarkon ways is a lot more fluid than mine, theres a alot of room for mistake like you saw me do.
\[ u=3x-6\]

Answer 16

well you can if you want to i wouldnt mind seeing another example but if not its kool too bnut

Answer 17

\[(u+6)^3\div3^3=x^3\] do you agree up tho here?

Answer 18

yosh

Answer 19

and the du/3=dx and you have all your subs in there so your integral would look like this\[1/27\int\limits_{}^{}((u+6)^3/u)du \]

then youll get a bunch od u's cancelling and the last term should be a CONSTANT which is where the ln(u) comes into play. try it out

Answer 20

ahhh just tried it out, neat!

Answer 21

but the other way would be easier, but calculus is def on how creative you can be when doing integrals, der, or any apps good luck with the class

Answer 22

yah i see what you mean bnut, thanks for your help as well XD