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

• ae4jm
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.
ae4jm
[SOLVED] Find integral of sqrt((1-x)/(1+x))

## Homework Statement

$$\int\sqrt{\frac{1-x}{1+x}}dx$$

## The Attempt at a Solution

I have started by multiplying by $$\sqrt{1-x}$$ in the numerator and denominator. Then I separated the two fractions to get
$$\int\frac{1}{1-x^2}dx$$$$-\int\frac{x}{1-x^2}dx$$ I'm stuck here! Any help is greatly appreciated!

Well the first one can be solved by partial fractions (among other methods) and the second one can be solved by a simple substitution.

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.

The denominator should be sqrt(1-x^2) instead of just 1-x^2, correct? My easy algebra mistake!

Right... you can still solve with a trig sub, and a regular sub.

Thanks to all, does this look correct?

This is what I got:

$$=sin^-^1x+2\sqrt{1-x^2}+c$$

Thanks to everyone!

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.

I believe my 2 should have canceled out with the $$\frac{1}{2}$$

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

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!

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:

## 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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