Integrating Shell Collapse Velocity Equation for Free-Fall Time Calculation

[TheTourist]

I have derived the following equation for the velocity of a collapsing shell of a collapsing star.

$$\frac{dr}{dt}$$=-[GM($$\frac{1}{2}$$-$$\frac{1}{R}$$)]^1/2​

I now need to integrate this to find the free-fall time, and the hint is to manipulate the equation into a form where you can use a standard integral: substitute x=r/R and then use the standard integral $$\int$$[$$\frac{x}{1-x}]$$^1/2 dx=$$\pi$$/2

I am completely stuck on how to do this

 

[tiny-tim]

Hi TheTourist!

(have an integral: ∫ and a square-root: √ and a pi: π )

hmm … that RHS can't possibly be correct …

shouldn't it be a function of r? ​