Find integral of sqrt((1-x)/(1+x))

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In summary, the conversation was about finding the integral of sqrt((1-x)/(1+x)), which was solved by multiplying by sqrt(1-x) in the numerator and denominator and then separating the fractions. The first integral can be solved using partial fractions and the second one can be solved using a substitution. After some help and corrections, the final answer was found to be sin^-1x+sqrt(1-x^2)+c.
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[SOLVED] Find integral of sqrt((1-x)/(1+x))

Homework Statement


[tex]\int\sqrt{\frac{1-x}{1+x}}dx[/tex]


Homework Equations





The Attempt at a Solution



I have started by multiplying by [tex]\sqrt{1-x}[/tex] in the numerator and denominator. Then I separated the two fractions to get
[tex]\int\frac{1}{1-x^2}dx[/tex][tex] -\int\frac{x}{1-x^2}dx[/tex] I'm stuck here! Any help is greatly appreciated!
 
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  • #2
Well the first one can be solved by partial fractions (among other methods) and the second one can be solved by a simple substitution.
 
  • #3
Please check your multiplication for the denominator. Although wrong, the integrals you have are also easily integrable, first term with trigonometric substitution and the second term with ordinary variable substitution.
 
  • #4
The denominator should be sqrt(1-x^2) instead of just 1-x^2, correct? My easy algebra mistake!
 
  • #5
Right... you can still solve with a trig sub, and a regular sub.
 
  • #6
Thanks to all, does this look correct?

This is what I got:

[tex]=sin^-^1x+2\sqrt{1-x^2}+c[/tex]

Thanks to everyone!
 
  • #7
You are almost there. It's always a good idea to double check your integration by computing the derivative your result. If you do so, you will see you have a slight mistake.
 
  • #8
I believe my 2 should have canceled out with the [tex]\frac{1}{2}[/tex]

I'm now getting the answer [tex]=sin^-^1x+\sqrt{1-x^2}+c[/tex]

After following your help, I did get the integral that I started with after I split it into two fractions. Thanks for your help and for catching my mistake!
 
  • #9
Ahh. Much better.

EDIT:
Now that you know what the answer is supposed to be, you should go back to your derivation to see where you dropped a factor of 1/2 (or added a factor of 2).
 
Last edited:

Related to Find integral of sqrt((1-x)/(1+x))

What is the function being integrated?

The function being integrated is sqrt((1-x)/(1+x)).

What is the domain of the function?

The domain of the function is all real numbers except for x = -1.

What is the proper notation for the integral?

The proper notation for the integral is ∫ sqrt((1-x)/(1+x)) dx.

What is the technique used to solve this integral?

The technique used to solve this integral is substitution, where u = 1+x and du = dx.

What is the final result of the integral?

The final result of the integral is -2√(1-x)/(1+x) + C.

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