📜  门| GATE 2017 MOCK II |问题 29

📅  最后修改于: 2021-09-27 05:36:03             🧑  作者: Mango

以下 Integral 的值为:

cos
(A) π
(B) π
(C) π/2
(D)答案: (C)
解释:
Use  \int_{a}^{b}f(x)dx = \int_{a}^{b}f(a + b-x)dx \\ \therefore I=\int_{-\pi }^{\pi }\frac{cos^{2}x}{1+a^{x}}dx, a+b=0 \\ I=\int_{-\pi }^{\pi }\frac{cos^{2}(0-x)}{1+a^{(0-x)}}dx=\int_{-\pi }^{\pi }\frac{a^{x}cos^{2}x}{1+a^{x}}dx \\ 2I=\int_{-\pi }^{\pi }\frac{1+a^{x}}{1+a^{x}}cos^{2}xdx=2\int_{0 }^{\pi }cos^{2}xdx \\ I = 2\int_{0}^{\pi/2 }cos^{2}xdx=2.\frac{1}{2}.\frac{\pi }{2}=\frac{\pi }{2}这个问题的测验