Question

The substitution used is u= , formula number .

After substitution, ∫sqrte^8x−25dx=∫ ? du.

Answer 1

\[\int\limits (\sqrt{e^{8x}}-25 ) dx \\ \text{ or } \int\limits \sqrt{e^{8x}-25} dx?\]

Answer 2

the 2nd one

Answer 3

is this fill in the blank
or did you mean to put u= something?

Answer 4

fill in the blank

Answer 5

ok so we do want to integrate this thing then...
hmmm... I'm thinking of a trig sub...(not an algebraic one)
let's play with this...
\[\int\limits \sqrt{(e^{4x})^2-25} dx \\ \sqrt{25} \int\limits \sqrt{ \frac{1}{25} (e^{4x})^2-1} dx \\ 5 \int\limits \sqrt{(\frac{e^{4x}}{5})^2-1} dx \\ \text{ recall } \tan^2(\theta)=\sec^2(\theta)-1 \\ \text{ so \let } \sec(\theta)=\frac{e^{4x}}{5} \\ \]

Answer 6

differentiate both sides

Answer 7

what did you use as your u?

Answer 8

nothing
you need a trig sub to do this one

Answer 9

well you could replace theta with u if you want to

Answer 10

\[\sec(u)=\frac{e^{4x}}{5}\]

Answer 11

still need to differentiate both sides

Answer 12

if you want you can solve that for u

Answer 13

to fill in that space...

Answer 14

Its asking for a u though

Answer 15

I already replaced theta with u...

Answer 16

and then I just said you could solve that for u if you want

Answer 17

so that you can fill in that blank

Answer 18

I will solve for u if you need...
\[\sec(u)=\frac{e^{4x}}{5} \\ u=arcsec(\frac{e^{4x}}{5}) \]

could you try differentiate sec(u)=e^(4x)/5

Answer 19

u=e^(4x)

Answer 20

if you do that as u
you will still need to make another substitution later

Answer 21

sec(u)=e^(4x)/5 gets all the substitutions out of the way
so the next step is just to integrate

Answer 22

it will be a pretty easy integral by the way

Answer 23

if you are using a table to integrate you have not made that very clear
your table is not universal

Answer 24

yeah, sorry I am having to use tables to integrate. I have the whole integrate. I just needed the fill in the blanks I had posted

Answer 25

your table is not universal though
can you post a copy of your table

Answer 26

ok you suggested earlier that we do u=e^(4x)
what happens when you find du

Answer 27

du would be 4e^4xdx

Answer 28

\[\int\limits \sqrt{(e^{4x})^2-5^2} dx \\ \text{ Let } u=e^{4x} \\ du=4 e^{4x} dx \\ \]

guess what e^(4x) can be written as u (the one in our du)

Answer 29

\[u=e^{4x} \\ du=4 u dx \\ \frac{1}{4} \frac{1}{u} du=dx\]

Answer 30

so what does our integral look like in terms of u

Answer 31

it should make it clear what formula number you want to use after that

Answer 32

\[(1/4u)\sqrt{u^2-25}\]

Answer 33

\[\frac{1}{4} \int\limits \frac{\sqrt{u^2-5^2}}{u} du\]

Answer 34

what number does this look like in your book

Answer 35

hint look at the ones between 39 and 42

Answer 36

your formula is in there somewhere

Answer 37

its 41, thank you

Answer 38

yep
replace the u with's e^(4x)
and a's with 5
and multiple that whole result by 1/4