Question

Integration.

Answer 1

\[\int\limits_{}^{}e^\sqrt[3]{x}\]

Answer 2

I have a bunch of tools I could use, i'm just not sure which to apply here?

Answer 3

start with u sub

Answer 4

then do by parts

Answer 5

put \[x ^{\frac{ 1 }{ 3 }}=y,x=y^3,dx=3y^2dy\]
\[\int\limits e \sqrt[3]{x}dx=3 \int\limits y^2e^ydy+c\]
integrate by parts

Answer 6

so my usub would be
u=x^(1/3) ?

Answer 7

yes

Answer 8

\[du=\frac{ 1 }{ x }x^{-2/3}\]

Answer 9

dx

Answer 10

oo yes that is what I meant.

Answer 11

but then what? o.0

Answer 12

\[\large\rm du=\frac{1}{3}x^{-2/3}dx\qquad\to\qquad 3x^{2/3}du=dx\]Or written like this,\[\large\rm 3(x^{1/3})^2du=dx\]Do you see how you can make use of your \(\large\rm u\) here?

Answer 13

I can just put
3(u)^2 ?

Answer 14

\[\int\limits_{}^{}e^u*3u^2du\]

Answer 15

Yessss, good.

Answer 16

Then for integration by parts we choose
\[u=e^u\]\[dv=3u^2du\]

Answer 17

hmm or maybe not because then we got another weird integration

Answer 18

We want to "destroy" the polynomial portion, right?
Break it down?

So that should be our u.

Wow you should have used a different letter for your first substitution.
I would recommend using M and N for your parts maybe.

Answer 19

haha ok
and..I forget the antiderrivative of e^u. is that just e^u?

Answer 20

yes :)

Answer 21

\[e^u*3u^2-\int\limits_{}^{}e^u*6udu\] then we must do it again?

Answer 22

Mmmm good.

Answer 23

ya, another round of parts.

Answer 24

\[e^u*3u^3-6e^u*u+6e^u\]

Answer 25

Would be the final answer? +C

Answer 26

Close to final answer, yes.
*cough cough*
Undo your sub.

Answer 27

oh my. The u from the very beginning o.o heh I forgot.

Answer 28

\[3e^\sqrt[3]{x}*(\sqrt[3]{x})^2-6e^\sqrt[3]{x}*\sqrt[3]{x}+6e^\sqrt[3]{x}+C\]oh..I could have simplified that a whole lot.

Answer 29

but is that it?

Answer 30

Yes, good job :)

Answer 31

Wolfram confirms:
https://www.wolframalpha.com/input/?i=integral+of+e%5E(x%5E1%2F3)
They did a little bit of fancy factoring,
that's why theirs looks a tad different.

Answer 32

yaaay! So for e, do you think it's usually wisest to start with a u sub? just typically

Answer 33

Bahh I dunno :)

Answer 34

haha ok :)

Answer 35

thanks!!