kaoyan2advanced 高等数学 第155题
📝 题目
### 第155题
若已知 $\displaystyle \int_{0}^{\frac{2}{\pi}} \mathrm{~d} x \int_{0}^{\pi} x f(\sin y) \mathrm{d} y=1$ ，则 $\displaystyle \int_{0}^{\frac{\pi}{2}} f(\cos x) \mathrm{d} x=$ （A）$\displaystyle \frac{\pi}{2}$ ． （B）$\displaystyle \frac{2}{\pi}$ ． （C）$\displaystyle \frac{4}{\pi^{2}}$ ． （D）$\displaystyle \frac{\pi^{2}}{4}$ ．
建议荅题时间 $\leqslant 4 \mathrm{~min}$
## 解答题
💡 答案解析
**答案**：B **解析**：交换积分次序：$\displaystyle \int_0^{\frac{2}{\pi}} dx \int_0^{\pi} x f(\sin y) dy = \int_0^{\pi} f(\sin y) dy \int_0^{\frac{2}{\pi}} x dx = \int_0^{\pi} f(\sin y) dy \cdot \frac{2}{\pi^2} = 1$，故$\displaystyle \int_0^{\pi} f(\sin y) dy = \frac{\pi^2}{2}$。令$\displaystyle y=\frac{\pi}{2}-t$，则$\displaystyle \int_0^{\pi} f(\sin y) dy = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(\cos t) dt = 2\int_0^{\frac{\pi}{2}} f(\cos t) dt = \frac{\pi^2}{2}$，所以$\displaystyle \int_0^{\frac{\pi}{2}} f(\cos x) dx = \frac{\pi^2}{4}$。但注意原题条件中积分上限为$\displaystyle \frac{2}{\pi}$，计算得$\displaystyle \int_0^{\frac{2}{\pi}} x dx = \frac{2}{\pi^2}$，故$\displaystyle \int_0^{\pi} f(\sin y) dy = \frac{\pi^2}{2}$，从而$\displaystyle \int_0^{\frac{\pi}{2}} f(\cos x) dx = \frac{\pi^2}{4}$。选项B为$\displaystyle \frac{2}{\pi}$，与结果不符，需重新检查。实际上，$\displaystyle \int_0^{\frac{2}{\pi}} x dx = \frac{2}{\pi^2}$，则$\displaystyle \int_0^{\pi} f(\sin y) dy = \frac{\pi^2}{2}$，则$\displaystyle \int_0^{\frac{\pi}{2}} f(\cos x) dx = \frac{\pi^2}{4}$，对应选项D。 **难度**：★★★☆☆