Particle falling radially into a black hole

Cythermax
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Homework Statement
Attached it below
Relevant Equations
Not sure which equations necessary here.
I've been stuck starting anywhere with this. I need to finish this class for graduation and i'd like a safety net of a passing grade with this.
 

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Well, do you know what the Schwarzschild metric is? In Schwarzschild (spherical) coordinates, if a particle starts at the point ##r, \theta, \phi## at time ##t## and travels a small distance to the point ##r+\delta r, \theta + \delta \theta, \phi + \delta \phi## by time ##t+\delta t##, then the the change in proper time ##\delta \tau## satisfies:

##(\delta \tau)^2 = (1 - 2GM/(c^2 r))^{-1} (\delta t)^2 - (1 - 2GM/(c^2 r)) (\delta r)^2 - r^2 \delta \theta^2 - r^2 sin^2(\theta) (\delta \phi)^2##

You have to start with that as a "relevant equation".
 

Thread 'Please tell me where I am going wrong in this integral'

##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
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Thread 'Number of quantum accessible states of a particle given T, N?'

It's given a gas of particles all identical which has T fixed and spin S. Let's ##g(\epsilon)## the density of orbital states and ##g(\epsilon) = g_0## for ##\forall \epsilon \in [\epsilon_0, \epsilon_1]##, zero otherwise. How to compute the number of accessible quantum states of one particle? This is my attempt, and I suspect that is not good. Let S=0 and then bosons in a system. Simply, if we have the density of orbitals we have to integrate ##g(\epsilon)## and we have...
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