If $X= \int_0^X dt$, why is it that: $X^a= a \int_0^X t^{a-1} dt$

Question

I'm solving some problems in measure and integration theory, and found this result I cannot explain from my basic understanding:

If $X= \int_0^X dt$, then, $X^a= a \int_0^X t^{a-1} dt$

Can somebody please explain to me what am I missing?.

The 'if' part is straight forward, but cannot explain the derivative of $X^a$ that shows up in the 'then' part.

Thanks folks.