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设f′′(x)在[a,b]上连续且非零,设f′(x)≥m>0, ∀x∈[a,b],证明:$$\left|\int _a^b \sin(f(x))\,dx\right|\leq \frac{4}{m}$$
证明(MSE)
存在 c∈(a,b):
\begin{align*}
\abs{\int_a^b \sin f(x) \, dx}
&= \abs{\int_a^b f'(x)\sin f(x) \cdot \frac 1{f'(x)}\, dx}\\
&\le \frac 1{f'(a)}\abs{\int_a^c f'(x)\sin f(x)\, dx} + \frac 1{f'(b)}\abs{\int_c^b f'(x)\sin f(x)\, dx}\\
&\le \frac 1m \abs{\int_{f(a)}^{f(c)} \sin u \, du} + \frac 1m
\abs{\int_{f(c)}^{f(b)}\sin u\, du}
\end{align*}设$\beta = \alpha + 2k\pi + \gamma$, $γ<2π$, $k∈\Bbb N$\begin{align*}
\abs{\int_\alpha^\beta \sin u \, du}
&= \abs{\int_0^{\gamma}\sin u \, du}\\
&\le \int_0^{\pi} \abs{\sin u}\, du\\
&= 2.
\end{align*}代入上面的不等式即证. |
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