证明下列不等式:
(1) 设 $x \in[0, \pi], t \in[0,1]$, 则 $\sin t x \geq t \sin x$;
(2) 设 $p>0$, 则 $\int_0^{\frac{\pi}{2}}|\sin u|^p \mathrm{~d} u \geq \frac{\pi}{2(p+1)}$;
(3) 设 $x \geq 0, p>0$, 则 $\int_0^x|\sin u|^p \mathrm{~d} u \geq \frac{x|\sin x|^p}{p+1}$.
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$\text{B.}$
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