How to prove the inequalities $\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}(\sin x)^ndx\ge 0$ 
Show that:
  $$\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}(\sin x)^ndx\ge 0$$

My idea:maybe $\sin{(x^n)}\ge (\sin{x})^n？$

Now  I have other solution,  since $$f(x)=\dfrac{\sin{x}}{x}$$ is Monotone decreasing in $x\in (0,1)$,
so
$$\dfrac{\sin{x}}{x}\le\dfrac{\sin{x^{n}}}{x^n}$$
so
$$x^{n-1}\sin{x}\le \sin{(x^n)}$$
since
$$\sin^n{x}=\sin^{n-1}{x}\cdot \sin{x}\le x^{n-1}\sin{x}<\sin{(x^n)}$$
