if $\hat{V}\left(t\right)$ is a vector function of $t$, find the indefinite integral $\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt$
To solve thi first i find for the integrand with matrix since it's in the form of cross product, so
$\hat{V}\times \frac{d^2\hat{V}}{dt^2}=\begin{pmatrix}\hat{i} & \hat{j} & \hat{k} \\V_x & V_y & V_z \\\frac{d^2V_x}{dt^2} & \frac{d^2V_y}{dt^2} & \frac{d^2V_z}{dt^2}\end{pmatrix}=\left(V_y\frac{d^2V_z}{dt^2}-V_z\frac{d^2V_y}{dt^2}\right)\hat{i}+\left(V_z\frac{d^2V_x}{dt^2}-V_x\frac{d^2V_z}{dt^2}\right)\hat{j}+\left(V_x\frac{d^2V_y}{dt^2}-V_y\frac{d^2V_x}{dt^2}\right)\hat{k}$
and then substitute this value into the integral, i get
$\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt=\left[\int\left(V_y\frac{d^2V_z}{dt^2}\right)dt-\int\left(V_z\frac{d^2V_y}{dt^2}\right)dt\right]\hat{i}+\left[\int\left(V_z\frac{d^2V_x}{dt^2}\right)dt-\int\left(V_x\frac{d^2V_z}{dt^2}\right)dt\right]\hat{j}+\left[\int\left(V_x\frac{d^2V_y}{dt^2}\right)dt-\int\left(V_y\frac{d^2V_x}{dt^2}\right)dt\right]\hat{k}$
What should i do next? how to evaluate each of these indefinite integrals?
Consider that
$$\frac{d}{dt} (\mathbf{v} \times\mathbf{\dot{v}} ) = \mathbf{\dot{v}} \times\mathbf{\dot{v}} +\mathbf{v} \times\mathbf{\ddot{v}} $$
What is the cross product of a vector and itself?