VC2.9

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We have {\frac  {d}{dt}}[r\times {\frac  {dr}{dt}}]={\frac  {dr}{dt}}\times {\frac  {dr}{dt}}+r\times {\frac  {d^{2}r}{dt^{2}}}\,

=r\times {\frac  {d^{2}r}{dt^{2}}}\, [Since {\frac  {dr}{dt}}\times {\frac  {dr}{dt}}=0\,]

Applying integral bothsides,

\int a\cdot [r\times {\frac  {d^{2}r}{dt^{2}}}]\,dt=r\times {\frac  {dr}{dt}}+c\, where c is an arbitrary constant vector.

Now \int a\cdot [r\cdot {\frac  {d^{2}r}{dt^{2}}}]\,dt=a\cdot \int r\times {\frac  {d^{2}r}{dt^{2}}}\,dt=a\cdot [r\times {\frac  {dr}{dt}}+c]=a\cdot r\times {\frac  {dr}{dt}}+d\,

where d(=a.c) is an arbitrary constant scalar.

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