An elementary-looking integral inequality

Question

This might seem a bit easy but I still like to ask it for pedagogical reasons.

QUESTION. Is this inequality true for non-negative integers $n$? $$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\geq\frac1{n+1}.$$

Answer 1

For two increasing functions $f,g$ we have $\int_0^1 fg\geqslant \int_0^1 f\cdot \int_0^1 g$ (Chebyshev's inequality). Apply this for $f(x)=x^n$ and $g(x)=\frac{\pi}2 \sin (\frac{\pi}2 x)$.

Answer 2

There is equality for $n=0$. For $n\geq 1$ use $\sin(\frac{\pi}{2}x)\geq x$ for $x\in [0,1]$ to obtain $$\frac{\pi}{2}\int_0^1 x^n\sin\left(\frac{\pi}{2}x\right) dx> \frac{3}{2}\int_{0}^1 x^{n+1}dx=\frac{3}{2(n+2)}\geq \frac{1}{n+1}.$$