VC2.8

$\int_1^2 r\times \frac{d^2 r}{dt^2}\,dt=[r\times\frac{dr}{dt}]\,$ at [1,2]

Given, $r(t)=5t^2i+tj-r^3k\,$

$\frac{dr}{dt}=10ti+j-3t^2k\,$

Therefore, $r\times \frac{dr}{dt}=(5t^2i+tj-r^3k)\times (10ti+j-3t^2k)\,$

= $\begin{vmatrix} i & j & k \\ 5t^2 & t & -t^3 \\ 10t & 1 & -3t^2\end{vmatrix}\,$

= $-2t^3i+5t^4j-5t^2k\,$

Therefore, $\int_1^2 r\times \frac{d^2 r}{dt^2}\,dt=[-2t^3i+5t^4j-5t^2k]\,$ at t=[1,2]

= $-2(8-1)i+5(16-1)j-5(4-1)k=-14i+75j-15k\,$