I want to check if one of the following is uniformly continuous and differentiable on $\mathbb{R}$.
- $\int_0^{x^3}\sin t\, dt$
- $\int_0^x\cos (t^3)\, dt$
- $\int_0^x[t]\, dt$
- $\int_0^xe^{t^3}\, dt$
To check that do we have to check if the derivative is Lipschitz continuous?
If yes I suppose that teh second one satisfies that property, since the cosine function is bounded.
Your answer for 2) is correct.
For 1) note that the function is just $1-\cos (x^{3})$. Can you show that this is not uniformly continuous? [Consider the points $(2n\pi)^{1/3}$ and $(2n\pi+\frac {\pi} 2)^{1/3}$.
For 3)use the fact that you can compute the integral explicitly by splitting the integral into $0$ to $1$, $1$ to $2$ etc. Now check that the values at $n$ and $n+\frac 1 n$ difer by $1$ which shows that the function is not uniformly continuous. I will leave differentiabilty to you.
For 4) here is a hint: $0<x<y$ implies $\int_x^{y} e^{t^{2}} dt \geq e^{x^{2}} (y-x)$. Choose $x$ and $y$ appropriately to contradict uniformly continuity.