Question

More Indefinite Integrals !

Indefinite Integral 3 sq root (x^2) (dx)

I have no idea what 3sq root (x^2) is?

Help is appreciated!

Answer 1

First help me rewrite the integral...

\[\sqrt[3]{x^2}\] (dx)

Answer 2

\[\Large\rm \int\limits \sqrt[3]{x^2}dx\]Do you mean like this? cube root?
Ah ok yes.

Answer 3

yes !

Answer 4

We want to rewrite our root as a rational exponent.
Example:\[\Large\rm \sqrt[5]{x^3}=x^{3/5}\]The `numerator` represents the degree of the `power` on x.
The `denominator` represents the degree of the `root`.

Answer 5

so that means...\[\sqrt[3]{x^2} = x^{2/3}\]

Answer 6

Mmm good good good.

Answer 7

And you apply your power rule to integrate it.

Answer 8

ok! great, thank you @zepdrix :)

Answer 9

yay team \c:/

Answer 10

my final answer is \[\frac{ 3 }{ 5 } x ^{5/3}\]

Answer 11

So when you divided by the new exponent, it flipped?
Yah looks good!

Answer 12

yes ! this is the new rule I learned ? Its because I want to simplify the integral

Answer 13

@zepdrix can you help me with this one on how they got -5?

\[\frac{ 1 }{ x^5 } (dx)\]

Answer 14

Jaza Jaza Jaza -_-
Oh boy, you really need to brush up on your exponent rules!

Answer 15

Yes I do! >.< that's why I'm here!

Answer 16

\[\Large\rm x^{-a}=\frac{1}{x^a}\]The negative tells us to take the reciprocal.
Examples:\[\Large\rm \frac{1}{x^3}=x^{-3}\]
\[\Large\rm \frac{1}{x^{-2}}=x^2\]

Answer 17

You're only doing this to the x though.
So here is another important example:\[\Large\rm \frac{2}{x^5}=2x^{-5}\]So when I say reciprocal, I mean the 1/x^5 portion.
Notice that the 2 still stays on top.

Answer 18

I see!! That's simply put! Thank you :)

Answer 19

so that means when I integrate this I will get

\[x ^{-5+1} = x ^{-4}\]

and then...\[\frac{ x^{-4} }{ 4 } \]

Answer 20

is this right?

Answer 21

\[\Large\rm \int\limits x^{-5}dx=\frac{x^{-4}}{-4}\]Woops, missing a minus sign there.

Answer 22

Ok, so now I know the minus sign doesn't "go away" if need be in the denominator

Answer 23

Yah, if you wanted to, you could rewrite it in the form it was given to you.\[\Large\rm \frac{x^{-4}}{-4}=\frac{1}{-4x^4}\]To remove the negative it just switches places.
If it's in the numerator, it goes to the denominator (as in this example).
If it was in the denominator, it would move to the numerator position.

Answer 24

Or were you talking about the other minus sign?

Answer 25

the minus sign, but in the previous response you answered my next question! :)

Answer 26

It doesn't really matter `where` you put the negative sign.
It's just a negative one, you can float it around anywhere.\[\Large\rm \frac{-1}{2}=\frac{1}{-2}=-\frac{1}{2}\]If that's what your question is about.
So I would write the answer like this probably:

Answer 27

\[\Large\rm =-\frac{1}{4x^4}\]

Answer 28

Oh okay!! Got it, wow thank you @zepdrix