$
U(P) 
&= - G integral_(r=0)^R integral_(theta = 0)^(pi) integral_(phi=0)^(2 pi) rho(r)/q r^2 dif r dif theta sin(theta) dif phi
&"on choisit de prendre" phi in [0,2 pi[, theta in [0, pi[\
&=- 2 pi G integral_(r=0)^R integral_(theta = 0)^(pi) rho(r)/q r^2 dif r dif theta sin(theta) &"intégration par rapport à" phi "indépendant de" rho\
&=- 2 pi G integral_(r=0)^R integral_(theta = 0)^(pi) rho(r)/q r^2 dif r dif theta sin(theta) &"intégration par rapport à" phi "indépendant de" rho\
&=- 2 pi G integral_(r=0)^R integral_(theta = 0)^(pi) rho(r)/q 11 r^2 dif r dif theta sin(theta) &"intégration par rapport à" phi "indépendant de" rho\
&= - 2 pi G integral_(r=0)^R integral_(theta = 0)^(pi) rho(r)/(r^2 + s^2 - 2 r s cos(theta))^(1\/2) r^2 dif r dif theta
&"formule" q^2 = r^2 + s^2 - 2 r s cos(theta) quad (1)\     
&= - 2 pi G 
integral_(r=0)^R 
integral_(u=-1)^1 
(rho(r))/((r^2 + s^2 - 2 r s u)^(1\/2)) r^2 dif r dif u 
&u = cos(theta), dif u = - sin(theta) d theta\
&= 
- 2 pi G integral_(r=0)^R rho(r) r^2 integral_(u=-1)^1 (dif u)/(((r^2 + s^2) + (-2 r s ) u )^(1\/2)) dif r
&integral (dif x)/sqrt(a + b x) = (2 sqrt(a + b x))/b\
&= - 2 pi G integral_(r=0)^R rho(r) r^2 [2 sqrt(r^2 + s^2 - 2 r s u)/(- 2 r s)]_(-1)^1 dif r\
&= - 2 pi G integral_(r=0)^R rho(r) r^2 1/ (r s) (sqrt(r^2 + s^2 + 2 r s) - sqrt(r^2 + s^2 - 2 r s)) dif r\
&= - (2 pi G)/s integral_(r=0)^R rho(r) r (sqrt((r+s)^2) - sqrt((r-s)^2)) dif r\
&= - (2 pi G)/s
integral_(r=0)^R rho(r) r  ((r +s) - (s-r)) dif r
& sqrt((r-s)^2) = abs(r-s) = s-r "car" r < s\
&= - (2 pi G)/s integral_(r=0)^R rho(r) r (2 r) dif r\
&= - (2 pi G)/s integral_(r=0)^R rho(r) 2 r^2 dif r\
&= - G/s integral_(r=0)^R rho(r) 4 pi r^2 dif r\
&= - (G M) /s
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