Integral x^5*sqrt(9 + x^3)

[SpeedyG]

Integral [x^5*sqrt(9 + x^3)]dx
Can someone just get me started or tell me what method to use? I'm sure this is easier than I'm making it, but I have no clue how to start. I've tried several different ways but so far none of them are working. But then again I might be doing those wrong...

 

[Deleted member 4993]

Integral [x^5*sqrt(9 + x^3)]dx
Can someone just get me started or tell me what method to use? I'm sure this is easier than I'm making it, but I have no clue how to start. I've tried several different ways but so far none of them are working. But then again I might be doing those wrong...

Start with integration by parts.

du = (9/4) * x² * √(9+x³)dx → u = (9 + x³)^3/2

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/threads/77972-Read-Before-Posting!!

 

[pka]

Integral [x^5*sqrt(9 + x^3)]dx
Can someone just get me started or tell me what method to use?

If you use \(\displaystyle u=9+x^3\) then \(\displaystyle du=3x^2dx\) from which we can get
\(\displaystyle \displaystyle\large\int {(u - 9)\sqrt u \frac{{du}}{3}} \)

 

[Deleted member 4993]

If you use \(\displaystyle u=9+x^3\) then \(\displaystyle du=3x^2dx\) from which we can get
\(\displaystyle \displaystyle\large\int {(u - 9)\sqrt u \frac{{du}}{3}} \)

I thought you did not like substitution!!!

 

[pka]

I thought you did not like substitution!!!

You are quite right. I do not like it being used before it is needed.
In this case, there is really no other way to do this anti-derivative.
My real fear is that the use of substitution will become an end-in-itself.
It is far far more important to know why find an anti-derivative than it is to know how.

 

[Prove It]

You are quite right. I do not like it being used before it is needed.
In this case, there is really no other way to do this anti-derivative.
My real fear is that the use of substitution will become an end-in-itself.
It is far far more important to know why find an anti-derivative than it is to know how.

Interesting. How do you apply the chain rule when taking a derivative of a composition?

 

[stapel]

Integral [x^5*sqrt(9 + x^3)]dx

So I guess the book you promoted here doesn't help with integration stuff...?

I have no clue how to start. I've tried several different ways...

If you're "tried several different ways" to start this, then you must have had some "clue how to start it"!

What did you get when you applied the "start" provided by the first or second reply here? Where have you bogged down? Please be complete. Thank you!

 

[lookagain]

SpeedyG, Subhotosh Khan brought up integration by parts.


If you choose integration by parts, it's

\(\displaystyle \int u*dv \ =\ uv \ -\ \int v*du\)


Rewrite

\(\displaystyle \int x^5\sqrt{9 + x^3} \ dx \ \ \ \) as

\(\displaystyle \int x^3*x^2(9 + x^3)^{\tfrac{1}{2}} \ dx \ \ \ \)


Let \(\displaystyle \ u \ = \ x^3, \ \ and \ \ let \)

\(\displaystyle dv \ = \ x^2(9 + x^3)^{\tfrac{1}{2}} \ dx. \)

 

[SpeedyG]

So I guess the book you promoted here doesn't help with integration stuff...?


If you're "tried several different ways" to start this, then you must have had some "clue how to start it"!

What did you get when you applied the "start" provided by the first or second reply here? Where have you bogged down? Please be complete. Thank you!

The FLCT helps with overarching concepts in calculus.
I tried doing a u-sub and what my teacher calls an ugly u-sub but neither worked so I guess what I really meant is that I had no idea how to start it correctly...
I think I have it pretty much figured out now. Thanks for all the help.