[裴礼文数学分析中的典型问题与方法习题参考解答]4.3.6

$f(x)$ 在 $[a,b]$ 上可导, $f'(x)\searrow$, $|f'(x)|\geq m>0$, 试证:

$$\bex \sev{\int_a^b \cos f(x)\rd x}\leq \frac{2}{m}. \eex$$

 

证明: 由换元法及积分第二中值定理,

$$\beex \bea \int_a^b \cos f(x)\rd x &=\int_{f(a)}^{f(b)} \frac{\cos y\rd y}{f'(f^{-1}(y))}\\ &=\frac{1}{f'(a)}\int_{f(a)}^\xi\cos y\rd y +\frac{1}{f'(b)}\int_\xi^{f(b)}\cos y\rd y\\ &=\frac{\sin\xi -\sin f(a)}{f'(a)} +\frac{\sin f(b)-\sin \xi}{f'(b)}\\ &\equiv I_1+I_2. \eea \eeex$$ 若 $I_1\cdot I_2\geq 0$, 则 $$\bex \sev{\int_a^b \cos f(x)\rd x} \leq \frac{\sev{\sin f(a)-\sin f(b)}}{f'(b)} \leq\frac{2}{m}; \eex$$ 若 $I_1\cdot I_2<0$, 则 $$\bex \sev{\int_a^b \cos f(x)\rd x} \leq \max\sed{\frac{\sev{\sin \xi-\sin f(a)}}{f'(a)},\frac{\sev{\sin f(b)-\sin \xi}}{f'(b)}} \leq\frac{2}{m}. \eex$$