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

[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!

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nicksauce
Homework Helper
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!

nicksauce
Homework Helper
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!

D H
Staff Emeritus
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 cancelled 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!

D H
Staff Emeritus