Can You Integrate x/sqrt(x(1-x))?

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Discussion Overview

The discussion revolves around the integral of the function x/sqrt(x(1-x)), exploring whether it can be computed and discussing various methods for approaching the problem. The scope includes mathematical reasoning and potential techniques for integration.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • Some participants express uncertainty about the possibility of integrating the function x/sqrt(x(1-x)).
  • One participant mentions that it is possible to find the integral using an integral table, suggesting that this method can save time.
  • Another participant requests clarification on what an integral table is and how it can be used.
  • A participant explains that an integral table lists integrals that have already been solved, which can help in finding solutions or methods for integration.
  • One participant suggests completing the square on the term inside the square root and using a trigonometric substitution as a typical approach to solve the integral.
  • A later reply provides a specific expression for the integral, but it is unclear how this result was derived or if it is universally accepted.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for integrating the function, and multiple approaches are discussed without resolution.

Contextual Notes

Some participants reference the use of integral tables and trigonometric substitutions, but the discussion does not clarify the assumptions or specific methods needed to arrive at the proposed solutions.

peace-Econ
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sorry, is it possible to take integral from this: x/sqrt(x(1-x))

Need help! Thanks!
 
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peace-Econ said:
sorry, is it possible to take integral from this: x/sqrt(x(1-x))

Need help! Thanks!

I believe it is possible. This is based on a quick look in an integral table.
 
Thank you for replying, but sorry, what is integral table? could you show that...?
 
peace-Econ said:
Thank you for replying, but sorry, what is integral table? could you show that...?

You are welcome.

An integral table is just a listing of the integrals already worked our for you. Sometimes you can find your integral in a table and this tells you that it is possible to work out a solution on your own. Also, seeing the answer can help you figure out how to solve it. So, basically this is a time saver to make sure you don't waste your time down a hopeless path.

This is the old way. The youngsters like to use the various computer based symbolic processors (Mathematical, Maxima, Maple, Wolfram, etc.), but I'm kind of old fashioned. Still, if I don't find it in a table, I will try these good tools.

Anyway, I've attached a PDF of the page from my table and you can see that numbers 163 and 164 will give you the answer. If you are just trying to get the answer, then you are all set, however if you want to work out the method, you need to figure it out. Looking at the table, it's not hard to see how they figured this out. Number 164 can be found by adding a term that forces the integral to have a du/sqrt(u) form. Of course, if you add a term, then you have to subtract it also, and you get the additional integral. Number 163 gives you the clue to solve by making the appropriate substitution that will give the inverse sine solution.
 

Attachments

The typical way for dealing with this type of integral is to complete the square on the term inside the square root in the denominator. Once you do that, make the appropriate trig substitution and go from there.
 
Thank you so much for your help!
 
int(x/sqrt(x(1-x))) = 0.5(atan((x-0.5)/sqrt(-x(x-1))) - sqrt(x-x^2) + C
 

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