How to Integrate (x) sq. root of ((x^2) - 1)?

  • Thread starter Prototype
  • Start date
  • Tags
In summary: No worries, Jameson. Thanks for catching it! In summary, the conversation is discussing the best way to solve for the square root of x^2-1 using substitution. Two methods are suggested, one using u = x^2-1 and the other using x = \cosh t. Both methods involve taking the derivative of the inside function and then substituting it into the original equation.
  • #1
(x) sq. root of ((x^2) - 1)?
Physics news on
  • #2
notice that the derivative of the inside is the same power as the one outside...

hmm, I'm thinking substitution... lol

let u = x^2 - 1
du = 2x dx
dx = du/2x

therefore you will then get (in terms of u)

1/2* (integral of sqrt(u))


or somethign like that..

hope you can do the rest

PS, is this an ap calc question?
Last edited:
  • #3
I would use the wonderful substiution:

[tex]x = \sec (u)[/tex]
  • #4
Or you could notate like this...

[tex]u = x^2-1[/tex]

[tex]du = 2x dx[/tex]

[tex]\frac{du}{2} = x dx[/tex]
Last edited by a moderator:
  • #5
Jameson said:
That last step is a typo or it's incorrect.

It should be:

[tex]u = x^2-1[/tex]

[tex]du = 2x dx[/tex]

[tex]\frac{du}{2} = x dx[/tex]

Your end substitution is correct, with [tex]\frac{1}{2}\int\sqrt{u}du[/tex]
but that one step is off.

can you explain why it is wrong?

i only know through calc bc, got the ap tommorow... if that explains it or not...

it would make sense to let dx = du/2x, so that i could just substitute it in

can't i treat dx and x as separate variables?
  • #6
I believe Jameson is just worried about your notation. du/(2x)=dx would be okay.

As well, Zurtex' method will work, but it's not necessary as the other suggestion in this thread is easier.
Last edited:
  • #7
It's not wrong, either way works:

xdx= du/2 so that [tex]x\sqrt{x^2-1)[/tex] becomes u1/2du/2 or

dx= du/2x so that [tex]x\sqrt{x^2-1}[/tex] becomes xu1/2du/2x and the x's cancel.
  • #8
I would suggest the delicious substitution

[tex]x=\cosh t [/tex]

  • #9
Ah, sorry, your notation does work. It just takes one extra step of cancelling the x's. I wasn't paying close enough attention. I fixed my post. Sorry for the messup.


Related to How to Integrate (x) sq. root of ((x^2) - 1)?

1. How do I integrate this into my experiment?

To integrate something into your experiment, you need to first understand what it is you are trying to integrate and how it relates to your experiment. Once you have a clear understanding, you can then determine the best way to incorporate it into your methods or data analysis. It may require adjusting your experimental design or using a specific statistical technique. It is important to carefully plan and document the integration process to ensure accurate and reliable results.

2. What are the potential challenges of integrating this?

Integrating something into your experiment can pose several challenges. These may include technical difficulties, compatibility issues, and potential biases. It is important to carefully consider these challenges and plan for them in advance. This may involve seeking guidance from experts in the field or conducting pilot studies to troubleshoot any potential issues.

3. How will integrating this impact my results?

Integrating something into your experiment can have a significant impact on your results. It may provide additional insights or change the interpretation of your findings. It is important to carefully consider the implications of the integration and how it may affect the overall conclusions of your study. It may also be helpful to compare your results with and without the integration to determine its impact.

4. Are there any ethical considerations when integrating this?

Depending on what you are integrating, there may be ethical considerations to take into account. This could include informed consent, potential harm to participants, or conflicts of interest. It is essential to carefully consider these ethical implications and ensure that your integration process follows ethical guidelines and regulations.

5. How can I ensure the validity and reliability of the integration process?

To ensure the validity and reliability of the integration process, it is important to carefully plan and document each step. This includes clearly defining the integration process, conducting thorough research and background checks, and using appropriate statistical methods. It is also important to conduct a sensitivity analysis to assess the impact of the integration on your results. Additionally, peer review and replication studies can help validate the integration process and its impact on the results.

Similar threads