Using Symmetry

Symmetry of the integrand often cuts work in half.

偶函数的积分

Even function property: if $f(x)$ is even ( $f(-x)=f(x)$ ), then $\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx.$

Proof: use interval additivity and evenness.

Odd function property: if $f(x)$ is odd ( $f(-x)=-f(x)$ ), then $\int_{-a}^a f(x)\,dx = 0.$

Proof: use interval additivity and oddness.

例 1 $\int_{-1}^1 x^2 dx = 2\int_0^1 x^2 dx = \tfrac{2}{3}$ .
例 2 $\int_{-2}^2 x^3 dx = 0$ .

利用对称性计算 $\int_{-2}^2 x^4 dx$ 。

参考答案

思路：偶函数，两倍 $[0,2]$ 。
答案： $2[\tfrac{x^5}{5}]_0^2 = \tfrac{64}{5}$ 。

符号	类型	读音/说明	在本文中的含义
$\int$	数学符号	integral	定积分符号