Question

Can someone help me find a solution to this integral?

(x^3) / sqrt((x^2) + 1)

Answer 1

the whole denominator is under the radical

Answer 2

\[\Large\rm \int\limits \frac{x^3}{\sqrt{x^2+1}}dx\]Ehhh you could take several different approaches to solve this one.
Trigonometric Substitution.
U-sub.
Integration by parts.

What do you prefer?

Answer 3

u-sub please:)

Answer 4

Umm ok I think we can make that work.
Let's borrow a couple powers of x from the numerator like this:\[\Large\rm \int\limits x^2\frac{x~dx}{\sqrt{x^2+1}}\]
We'll make the substitution: \(\Large\rm u=\sqrt{x^2+1}\)

Then taking a derivative gives us:\[\Large\rm du=\frac{x~dx}{\sqrt{x^2+1}}\]

Answer 5

can you show me more steps I can't quite understand how you got the final answer...

Answer 6

ah nvm I got it

Answer 7

\[\Large\rm \int\limits\limits x^2\frac{x~dx}{\sqrt{x^2+1}}=\int\limits x^2 du\]That takes care of a lot of the stuff for us.
More steps? Sure. I kinda rushed through that.

Answer 8

learning along :D

Answer 9

Where we having trouble? How I got from u to du?
Or why I chose that for u in the first place?

Answer 10

so all I need to do is solve for the integral of X^2 and get the integral plus c?

Answer 11

No no no.
We can't integrate x^2 with our differential du.
We have to also replace the x^2 with something in u.

Answer 12

@zepdrix how do you know what u to pick to get the answer the least painful way?

Answer 13

\[\Large\rm u=\sqrt{x^2+1}\]Square each side,\[\Large\rm u^2=x^2+1\]Subtract 1 from each side and voila!\[\Large\rm u^2-1=x^2\]

Answer 14

That takes some practice, just being able to eyeball it.
I had initially thought that \(\Large\rm u=x^2+1\) would be a good substitution.
But it leaves us with \(\Large\rm (stuff)^{3/2}\) in the numerator which would be difficult to deal with.

Answer 15

@study100 because when you do du it will give you that expression which is what we needed

Answer 16

I think Trig Sub is the best approach if you're scratching your head.
It probably ends up being a longer process.

But it's a safe way to get through the problem if you can't `see` the u substitution.
Sometimes they're hidden, like this one.

Answer 17

@zepdrix ohhh I see :) so that's why you leave the radical with u . Thank you!
@xapproachesinfinity Yep, but without knowing that in the first place, I guess I'd go with x^2 +1 and then find my way slowly to rad (x^2+1) , but it takes skills to get it in one go like zepdrix

Answer 18

Well I did it the wrong way on paper as my first approach lol :)
Took me a minute or two.

Answer 19

the first thing I thought about is rad(x^2+1)=u which is what he did here. but I made a mistake in differentiating that radical and got different result than he did. thus, I could see that x^2 separation to x.xdx

Answer 20

Where you at Johnny boy?
Confused still?

Answer 21

what if it was +1 instead of -1 under the radical

Answer 22

sorry -1

Answer 23

It would work out largely the same.
At least our du would be the same:\[\Large\rm \int\limits\limits x^2\frac{x~dx}{\sqrt{x^2+1}}=\int\limits\limits x^2~du\]Just the x^2 in front would change a lil bit.

Answer 24

\[\Large\rm \int\limits\limits\limits x^2\frac{x~dx}{\sqrt{x^2-1}}=\int\limits\limits\limits x^2~du\]Woops*

Answer 25

alrighty thanks for your help

Answer 26

It's a bit different for me then. Everyone has a different thought process and I need to pratctice some more with u substitution. I guess you're naturally good with this so that's really great for you :)

Answer 27

or you have lots of practice etc. Practice makes perfect.

Answer 28

well sometimes it works sometimes not and i got stuck as all people.

Answer 29

I actually i have gotten rusty in this stuff, i have been away from math. I'm recently trying to restore what i know LOL.

Answer 30

But of course practice is the key to this. it becomes second nature when you do it often lol

Answer 31

Oh you're still really great now! :) That's amazing.

Answer 32

Yes, I'm told that too lol :)

Answer 33

can you believe i'm starting over from precalc

Answer 34

Not really great, I make severe mistake in differentiation haha

Answer 35

oh? you're good at integration too, which is pretty important in Calc 1 and 2

Answer 36

Next semester i will get back to Calc1. my favorite integral are those that involve trigs

Answer 37

You're majoring in math? @xapproachesinfinity

Answer 38

Yes! and you

Answer 39

I'm just a calc student lol :)
I'm going to be in college soon, bio major
taking bio, prolly organic chem and calc 2

Answer 40

Oh! I see. Bio is an interesting major as well. Best of luck to you.
we meet again! I got a class now. My name is Hassan by the way

Answer 41

Nice meeting you, Hassan :) I'm Alice. Have fun in class!