How was the integral of x^{-1/2}*(1-x)^-1 derived?

[imurme8]

Claim: \int\frac{dx}{\sqrt{x}(1-x)}=\log{\frac{1+\sqrt{x}}{1-\sqrt{x}}}
Derivation confirms this, but how was this answer arrived at? IBP seems not to work, can't find a good u-substitution...

 

[Millennial]

What about the substitution x=u^2, followed by partial fraction decomposition?

 

[phyzguy]

Try writing:

\frac{1}{\sqrt{x}(1-x)} = \frac{1}{\sqrt{x}(1+\sqrt{x})(1-\sqrt{x})}

then use a u substitution and partial fractions

 

[imurme8]

Thanks guys!