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Integration in Mechanics

1. A integrand with a singular point: using a limiting process or excluding a small sphere [Jack, p.36, (1.36); p.37, l.-4-l.-1].



2. Evaluate the integral given in [Jack, p.60, (2.7)].
   Solution. Let x = cos g.  Then dx= -sin g dg.
   Let u = 1 + (a²/y²) and v = - (2a)/y.
   ò_{[0, p]} cos g sin g (1 + (a²/y²) - (2a cos g )/y)^-3 dg = ò_{[-1, 1]} x (u + vx)^-3 dx
   = v^-2 ò_{[u-v, u+v]} (z - u)z^-3dz (Let z = u + vx).



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