I'm struggling with the following problem (I'm in a Calc I class) - we've never covered this particular kind of integration question before, so any advice would be most welcome.

The problem asks:

> 
> $F\left(x\right)=\int \_0^{3x}\:\sqrt{t^2+2t}dt$ 
> 
> 
> Find $F'\left(1\right)$.
> 
> 
> 

Is F(x) distinct from f(x)? What would F'(x) be then?

The answer sheet I have proceeds with solving the problem using u-substitution. u = 3x, which makes u' (or du) = 3. I'm already a little lost, because I'm not accustomed to setting u to the boundaries (though I have rewritten the boundaries in terms of u for other problems).

The answer sheet ends up with:

$\int \_0^u\:\sqrt{t^2+2t}\:dt\:$

From there it ends up with:

$u'\sqrt{u^2+2u}$

From there it substitutes back the original value of u, from which we can solve F'(1). This step is maybe the most confusing, I'm not sure why u' is being multiplied against the function, and why u is being substituted for t in general. Any advice/insight?