Cross product of a vector with its second derivative.

Question

I'm currently learning vector calculus in my classical physics course, and am asked to evaluate the following integral.

$$\int{\mathbf A \times \mathbf A''} dt$$

Where $\mathbf A$ is some vector, and $\mathbf A''$ denotes the second derivative of the vector $\mathbf A$.

I'm not sure how to even begin to integrate this. I know that $ \mathbf A \times \mathbf A = 0$ , but I do not know how that translates to the second derivative.

Any help would be appreciated. Thank you.

Answer 1

Hint: $$({\bf A}\times{\bf A}')' = {\bf A}'\times{\bf A}' + {\bf A}\times{\bf A}'' = {\bf A}\times{\bf A}''$$