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 $n$?
> $$\frac{\pi}2\int\_0^1x^n\sin\left(\frac{\pi}2x\right)dx\geq\frac1{n+1}.$$
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An elementary-looking integral inequality
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 $g(x)=\frac{\pi}2 \sin (\frac{\pi}2 x)$.


