Question

Use the substitution method to evalute the integral ∫((2x^4) +1))^4 x^3 dx.
I managed to find u = 2x^4 + 1, then I found du = 8x^3 dx, and then du/8 = x^3 dx. I am totally baffled as to what the next step is?

Answer 1

You are so close!\[\int(2x^4+1)^4x^3dx\]substitute what you have and the magic occurs\[u=2x^4+1\]\[\frac{du}8=x^3dx\]

Answer 2

\[\int u^4(\frac{du}8)=\frac18\int u^4du\]

Answer 3

@tbreez712 does this make sense to you?

Answer 4

Sorry I was walking my dog. I'm a little confused as to what you did?

Answer 5

\[\int(2x^4+1)^4x^3dx\]\[u=2x^4+1\]\[\frac{du}8=x^3dx\]

Answer 6

yep yep

Answer 7

so all we have is a substitution
just make sure you recognize which part goes where
putting a few more parentheses may help:\[\int(2x^4+1)^4(x^3dx)\]\[u=2x^4+1\]\[\frac{du}8=x^3dx\]so substituting in terms of u we get\[\int(u^4)(\frac{du}8)\]looking reasonable?

Answer 8

yes

Answer 9

we can take the constants out of the integral, so this is\[\frac18\int u^4du\]you should be able to integrate that
what do you get?

Answer 10

so aall you did there was rewrite du/8 aas 1/8? And if i try to integrate that I would get u5/5?

Answer 11

I rewrote du/8 as du*(1/8) and took the 1/8 out of the integral sign
and yes, the integral itself is (u^5)/5
don't forget to keep the constant though!

Answer 12

sorry for sounding like a hillbilly, but is du the constant your talking about?

Answer 13

no the constant is the number 1/8
we can take constants out of the integral sign
it just makes it easier on the eyes\[\int2f(x)dx=2\int f(x)dx\]

Answer 14

ohhh ! i get it!

Answer 15

so 1/8 * u5/5?

Answer 16

u5/40 substituted back ian so (2x4+1)^5 / 40?

Answer 17

yep :)
nice job!

Answer 18

ahhh your awesome

Answer 19

+C

Answer 20

thanks for sticking through it with me!

Answer 21

aoh yeah forgot the +c

Answer 22

don't forget +C
and you're very welcome