V (r) = k*r^(n + 1)

v[u_]:=k/u^(n+1)

k=-10

l=3

mu=1

n=-2

fakev[u_]:=l^2/2/mu*u^2+v[u]

Plot[{v[1/r],l^2/mu/r^2,fakev[1/r]},{r,0,10},PlotRange->{-20,20}]

uequi=Solve[fakev'[uu]==0,uu]

fakev[uu]/.uequi

e=-5

Plot[2*mu*e-l^2/r^2-2*mu*v[1/r],{r,0,10}]

utp=NSolve[2*mu*e-l^2/r^2-2*mu*v[1/r]==0,r,Reals]

theta[rmax_]:=l*Integrate[1/Sqrt[2*mu*e-l^2*uu^2-2*mu*v[uu]],{uu,1/rmax,1/0.683772233983163}]

theta[1]*180/Pi

Plot[180/Pi*theta[rm],{rm,0.683772233983163,1.316227766016837}]

ParametricPlot[{{rm*Cos[theta[rm]],rm*Sin[theta[rm]]},{rm*Cos[-theta[rm]],rm*Sin[-theta[rm]]}},{rm,0.683772233983163,1.316227766016837}]