How to integrate these integrals from the free fall problem in classical mechanics?

Question

How to integrate: $\int_{0}^{2}\dot x^2(t) dt$ and $\int_{0}^{2} 2\dot x(t)dt$?

I would be very grateful for any answers!

Answer 1

I am guessing the dot means derivatve.
The second one is $$ \int_0^2 2 \dot{x}(t)\;dt = 2x(t)\bigg\vert_0^2 = 2x(2)-2x(0) $$ The reason is: $$ \frac{d}{dt}\;x(t) = \dot{x}(t) $$ so $x(t)$ is an anti-derivatvie of $\dot{x}(t)$.

For the first one here is no such simple evaluation.