奇偶函数的性质

深藏阁楼爱情的钟 2022-12-09 03:11 120阅读 0赞

g(x)为偶函数，h(x)为奇函数，那么有：

① g(x)=f(x)+f(-x)

② h(x)=f(x)-f(-x)

由①式+②式得出：

f(x)=1/2 \* ( g(x)+h(x) )

于是有：

![\\int\_\{-a\}^\{a\} f(x) dx = \\int\_\{-a\}^\{a\} \\frac\{1\}\{2\}(g(x)+h(x))dx = \\int\_\{0\}^\{a\}g(x)dx+0 = \\int\_\{0\}^\{a\}g(x)dx = \\int\_\{0\}^\{a\}\[f(x)+f(-x)\]dx][int_-a_a_ f_x_ dx _ _int_-a_a_ _frac_1_2_g_x_h_x_dx _ _int_0_a_g_x_dx_0 _ _int_0_a_g_x_dx _ _int_0_a_f_x_f_-x_dx]

例题：

求  ![\\int\_\{-1\}^\{1\}\\frac\{1\}\{1+e^\{\\frac\{1\}\{x\}\}\}dx][int_-1_1_frac_1_1_e_frac_1_x_dx] 的值

解：

![\\int\_\{-1\}^\{1\}\\frac\{1\}\{1+e^\{\\frac\{1\}\{x\}\}\}dx = \\int\_\{0\}^\{1\}\[\\frac\{1\}\{1+e^\{\\frac\{1\}\{x\}\}\}+\\frac\{1\}\{1+e^\{-\\frac\{1\}\{x\}\}\}\]dx = \\int\_\{0\}^\{1\}\[\\frac\{1\}\{1+e^\{\\frac\{1\}\{x\}\}\}+\\frac\{e^\{\\frac\{1\}\{x\}\}\}\{e^\{\\frac\{1\}\{x\}\}+1\}\]dx = \\int\_\{0\}^\{1\}1dx=1][int_-1_1_frac_1_1_e_frac_1_x_dx _ _int_0_1_frac_1_1_e_frac_1_x_frac_1_1_e_-_frac_1_x_dx _ _int_0_1_frac_1_1_e_frac_1_x_frac_e_frac_1_x_e_frac_1_x_1_dx _ _int_0_1_1dx_1]

[int_-a_a_ f_x_ dx _ _int_-a_a_ _frac_1_2_g_x_h_x_dx _ _int_0_a_g_x_dx_0 _ _int_0_a_g_x_dx _ _int_0_a_f_x_f_-x_dx]: https://private.codecogs.com/gif.latex?%5Cint_%7B-a%7D%5E%7Ba%7D%20f%28x%29%20dx%20%3D%20%5Cint_%7B-a%7D%5E%7Ba%7D%20%5Cfrac%7B1%7D%7B2%7D%28g%28x%29&plus;h%28x%29%29dx%20%3D%20%5Cint_%7B0%7D%5E%7Ba%7Dg%28x%29dx&plus;0%20%3D%20%5Cint_%7B0%7D%5E%7Ba%7Dg%28x%29dx%20%3D%20%5Cint_%7B0%7D%5E%7Ba%7D%5Bf%28x%29&plus;f%28-x%29%5Ddx
[int_-1_1_frac_1_1_e_frac_1_x_dx]: https://private.codecogs.com/gif.latex?%5Cint_%7B-1%7D%5E%7B1%7D%5Cfrac%7B1%7D%7B1&plus;e%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%7Ddx
[int_-1_1_frac_1_1_e_frac_1_x_dx _ _int_0_1_frac_1_1_e_frac_1_x_frac_1_1_e_-_frac_1_x_dx _ _int_0_1_frac_1_1_e_frac_1_x_frac_e_frac_1_x_e_frac_1_x_1_dx _ _int_0_1_1dx_1]: https://private.codecogs.com/gif.latex?%5Cint_%7B-1%7D%5E%7B1%7D%5Cfrac%7B1%7D%7B1&plus;e%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%7Ddx%20%3D%20%5Cint_%7B0%7D%5E%7B1%7D%5B%5Cfrac%7B1%7D%7B1&plus;e%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%7D&plus;%5Cfrac%7B1%7D%7B1&plus;e%5E%7B-%5Cfrac%7B1%7D%7Bx%7D%7D%7D%5Ddx%20%3D%20%5Cint_%7B0%7D%5E%7B1%7D%5B%5Cfrac%7B1%7D%7B1&plus;e%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%7D&plus;%5Cfrac%7Be%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%7D%7Be%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D&plus;1%7D%5Ddx%20%3D%20%5Cint_%7B0%7D%5E%7B1%7D1dx%3D1