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]

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

 

[shinobi20]

$$\frac{c\ dt}{r_s}=\frac{x^{3/2}}{1-x}dx=[-x^{1/2}+\frac{x^{1/2}}{1-x}]dx$$
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 ?