I'm solving a differential equation and have to integrate this term:
$\int \frac{dx(t)}{dt} x(t)^2 dt$
Partial integration gave me $0$ as result, so I gave it a try on wolframalpha. This came up with a solution that is analog to results I've seen for similar equations (The derivative just seems to equal to $1$ and we use a usual integration).
Wolframalphas solution: $\int \frac{dx(t)}{dt} x(t)^2 dt = \frac{x(t)^3}{3} + c$
My problem is, I can't follow the steps wolframalpha shows.
Why can you substitute like wolframalpha does? What are the rules used to do that?