VC2.7

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Given,r=2t^{2}i+tj-3t^{3}k\,

{\frac  {dr}{dt}}=4ti+j-9t^{2}k\,

{\frac  {d^{r}}{dt^{2}}}=4i+0j-18tk\,

Therefore, r\times {\frac  {d^{2}r}{dt^{2}}}=(2t^{2}i+tj-3t^{2}k)\times (4i+0j-18tk)\,

={\begin{vmatrix}i&j&k\\2t^{2}&t&-3t^{2}\\4&0&-18t\end{vmatrix}}\,

=-18t^{2}i-(-36t^{3}+12t^{3})j-4tk=-18t^{2}i+24t^{3}j-4tk\,

Hence,\int _{1}^{2}r\times {\frac  {d^{2}r}{dt^{2}}}\,dt=\int _{1}^{2}(-18t^{2}i+24t^{3}j-4tk)\,dt\,

=-18i\int _{1}^{2}t^{2}\,dt+24j\int _{1}^{2}t^{3}\,dt-4k\int _{1}^{2}t\,dt\,

=-18i[{\frac  {t^{3}}{3}}]+24j[{\frac  {t^{4}}{4}}]-4k[{\frac  {t^{2}}{2}}]\, at t=[1,2]

=-6(8-1)i+6(16-1)j-2(4-1)k=-42i+90j-6k\,

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