Schwarzschild coordinate time integral

[shinobi20]

Homework Statement

Integrate ##c dt = -\frac{1}{\sqrt{r*}} \frac{r^{3/2} dr}{r - r*}## to find the coordinate time as opposed to the proper time of an object falling into a Schwarzschild black hole.

Relevant Equations

##t## - coordinate time
##r## - radial coordinate
##r^*## - Schwarzschild radius (constant)

I have tried integration by parts where,

##c dt = -\frac{1}{\sqrt{r*}} \frac{r^{3/2} dr}{r - r*} = \frac{1}{\sqrt{(r*)^3}} \frac{r^{3/2} dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}##

##u = r^{3/2} \quad \quad dv = \frac{dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}##

##du = \frac{3}{2} r^{1/2} dr \quad \quad v = \tanh^{-1} \Big(\sqrt{\frac{r}{r*}} \Big)##

I think this is not the correct route.

 

[mitochan]

[tex]\frac{c\ dt}{r_s}=\frac{x^{3/2}}{1-x}dx=[-x^{1/2}+\frac{x^{1/2}}{1-x}]dx[/tex]
where ##x=\frac{r}{r_s}## and ##r_s## is Schwartzshild radius.
Integration seems easy.

 

[shinobi20]

[tex]\frac{c\ dt}{r_s}=\frac{x^{3/2}}{1-x}dx=[-x^{1/2}+\frac{x^{1/2}}{1-x}]dx[/tex]
where ##x=\frac{r}{r_s}## and ##r_s## is Schwartzshild radius.
Integration seems easy.

How do I integrate ##\frac{x^{1/2}}{1-x}##? I have done integration by parts but I can't find the answer.

*I could just use Mathematica but I want to learn how to deal with this kind of integral by hand.

 

[mitochan]

How about x=u^2 dx=2u du ?