Integration problem | Mathematics | Page 1

lusty umbra Jan 30, 2024, 1:44 AM

#

How on earth do I integrate this?

$\int x \left( x^2 + y^2 + z^2 \right)^{-\frac{5}{2}} ,dx$

Also its partial integration with respect to x.

grizzled canopyBOT Jan 30, 2024, 1:45 AM

#

lusty umbra How on earth do I integrate this? $\int x \left( x^2 + y^2 + z^2 \right)^{-\fr...

#

Wait patiently for a helper to come along.
Once someone helps you, say thank you and close the thread with:

+close

Feel free to nominate the person for helper of the week in #helper-nominations
Do not ping the mods, unless someone is breaking the rules.
If you're happy with the help you got here, and the server overall, you can contribute financially as well:

thorny hornetBOT Jan 30, 2024, 1:50 AM

#

dienophile

silk iron Jan 30, 2024, 1:56 AM

#

u substitution

#

do you know of this method?

#

first rewrite it it might help

#

$$\dfrac{x}{\left(x^2+z^2+y^2\right)^\frac{5}{2}}$$

thorny hornetBOT Jan 30, 2024, 1:57 AM

#

anthony

lusty umbra Jan 30, 2024, 2:11 AM

#

silk iron u substitution

Do you mean this?
$\int uv , dv = u \int v , dx - \int v \frac{du}{dx} , dx$

thorny hornetBOT Jan 30, 2024, 2:11 AM

#

dienophile

silk iron Jan 30, 2024, 2:13 AM

#

thats integration by parts

lusty umbra Jan 30, 2024, 2:15 AM

#

silk iron first rewrite it it might help

Okay but, what to do with the power? I know $\int \frac{1}{x^2+a^2} dx$ has this long weird formula, but how to approach this one? (Hopefully in a process thats not long winding 🙂 )

thorny hornetBOT Jan 30, 2024, 2:15 AM

#

dienophile

silk iron Jan 30, 2024, 2:15 AM

#

i can help in 15 minutes

#

a little busy atm

lusty umbra Jan 30, 2024, 2:15 AM

#

Okay sure

silk iron Jan 30, 2024, 2:24 AM

#

okay

#

so, do you know what U substitution is?

#

just yes or no will suffice

#

so what we will do is;

lusty umbra Jan 30, 2024, 2:26 AM

#

The process where you consider some f(x)=z and then differentiate it to substitute back?

silk iron Jan 30, 2024, 2:27 AM

#

let $$u = x^{2} + z^{2} + y^{2}$$

thorny hornetBOT Jan 30, 2024, 2:27 AM

#

anthony

silk iron Jan 30, 2024, 2:27 AM

#

so we have $$\dfrac{x}{\left(u\right)^\frac{5}{2}}$$

thorny hornetBOT Jan 30, 2024, 2:27 AM

#

anthony

silk iron Jan 30, 2024, 2:29 AM

#

we are going to take the derivative of $$u = x^{2} + z^{2} + y^{2}$$ with respect to x then rewrite it into terms of $$dx$$

thorny hornetBOT Jan 30, 2024, 2:30 AM

#

anthony

silk iron Jan 30, 2024, 2:30 AM

#

so what is the derivative of u = x^2 + z^2 + y^2

#

with respect to x

#

@lusty umbra

lusty umbra Jan 30, 2024, 2:31 AM

#

2x +xz²+xy²

silk iron Jan 30, 2024, 2:31 AM

#

no unfortunately that is not it

#

when we take the derivative of something that does not include our variable it is a constant

#

it becomes zero

#

so we have 2x + 0 + 0

#

= 2x

lusty umbra Jan 30, 2024, 2:32 AM

#

Sorry I integrated that oh well

#

Okay yes

silk iron Jan 30, 2024, 2:33 AM

#

so;

#

$$\frac{du}{dx} = 2x$$

thorny hornetBOT Jan 30, 2024, 2:33 AM

#

anthony

silk iron Jan 30, 2024, 2:33 AM

#

now can you rewrite that in terms of dx

lusty umbra Jan 30, 2024, 2:33 AM

#

du=2xdx

silk iron Jan 30, 2024, 2:34 AM

#

uh no

#

$$dx = \frac{1}{2x}$$

thorny hornetBOT Jan 30, 2024, 2:34 AM

#

anthony

silk iron Jan 30, 2024, 2:34 AM

#

or dx = du/2x

#

same thing

#

does that make sense to you?

lusty umbra Jan 30, 2024, 2:35 AM

#

thorny hornet **anthony**

Where did the du go here though?

silk iron Jan 30, 2024, 2:35 AM

#

the 1 is the du

#

you can say du/2x aswell its the samething as 1/2x

silk iron Jan 30, 2024, 2:36 AM

#

silk iron or dx = du/2x

which is what i mentioned here

lusty umbra Jan 30, 2024, 2:36 AM

#

Why is du=1 again? :-:

silk iron Jan 30, 2024, 2:36 AM

#

you might want to watch a organic chemistry tutor video

#

on the topic

#

he explains much better than i

#

(im just a student)

lusty umbra Jan 30, 2024, 2:37 AM

#

Okay is this topic u substitution?

silk iron Jan 30, 2024, 2:37 AM

#

yeah

#

i can walk you though it before you watch if you would like

#

we can finish off the question

lusty umbra Jan 30, 2024, 2:37 AM

#

Alright thanks will do

silk iron Jan 30, 2024, 2:38 AM

#

but its completely up to you

lusty umbra Jan 30, 2024, 2:38 AM

#

silk iron we can finish off the question

Yes please

silk iron Jan 30, 2024, 2:38 AM

#

okay

#

so right now we have

#

$$dx = \frac{1}{2x}$$

thorny hornetBOT Jan 30, 2024, 2:38 AM

#

anthony

silk iron Jan 30, 2024, 2:38 AM

#

we are going to sub that into our integral

#

sorry trying to figure out how u use integral sign in latex math notation

#

1 sec

#

$$(\frac{1]{2})\int(\frac{1}{u}^(\frac{5}{2})$$

thorny hornetBOT Jan 30, 2024, 2:41 AM

#

anthony
Compile Error! Click the errors reaction for more information.
(You may edit your message to recompile.)

silk iron Jan 30, 2024, 2:42 AM

#

thats what it becomes

#

the x cancels out with the x that was in the numerator

#

so we are left with 1/2 * our integral

#

now all we do is solve using power rule

#

we have $$u^\frac{-5}{2}$$

thorny hornetBOT Jan 30, 2024, 2:44 AM

#

anthony

silk iron Jan 30, 2024, 2:44 AM

#

apply power rule to that we get;

#

$$-\frac{2}{3u^\frac{3}{2}}$$

thorny hornetBOT Jan 30, 2024, 2:46 AM

#

anthony

silk iron Jan 30, 2024, 2:46 AM

#

we multiply this by our constant that was outside the integral

#

the 1/2

#

and we get $$-\frac{1}{3u^\frac{3}{2}}$$

thorny hornetBOT Jan 30, 2024, 2:46 AM

#

anthony

silk iron Jan 30, 2024, 2:46 AM

#

then we undo the substituion

#

we get

#

$$-\dfrac{1}{3\left(x^2+z^2+y^2\right)^\frac{3}{2}}$$

thorny hornetBOT Jan 30, 2024, 2:47 AM

#

anthony

silk iron Jan 30, 2024, 2:47 AM

#

final answer

#

c of course

#

i apologize for the terrible explanation

#

i hope you were able to somewhat follow

lusty umbra Jan 30, 2024, 2:48 AM

#

This was so simple
Dude thank you so much this was a lifesaver

silk iron Jan 30, 2024, 2:48 AM

#

if i were in call sharing my screen with my ipad it would have been much easier

silk iron Jan 30, 2024, 2:48 AM

#

lusty umbra This was so simple Dude thank you so much this was a lifesaver

im really glad you were able to follow

lusty umbra Jan 30, 2024, 2:49 AM

#

Yeah typing every message in latex isnt the best thing in the world

#

Thanks so much again

silk iron Jan 30, 2024, 2:49 AM

#

of course

#

id still advise you to watch a video on it

#

ill link one

#

https://www.youtube.com/watch?v=sdYdnpYn-1o

YouTube

The Organic Chemistry Tutor

How To Integrate Using U-Substitution

This calculus video tutorial provides a basic introduction into u-substitution. It explains how to integrate using u-substitution. You need to determine which part of the function to set equal to the u variable and you to find the derivative of u to get du and solve for dx. After replacing all x variables with u variables, find the antideriva...

▶ Play video

#

this helped me alot when i was taking this

lusty umbra Jan 30, 2024, 2:50 AM

#

silk iron Jan 30, 2024, 2:51 AM

#

@lusty umbra +close if that is all!

lusty umbra Jan 30, 2024, 2:51 AM

#

Yup

#

+close

#Integration problem