跳至内容
一、定义：
\(\begin{align} \int_{a}^{b} f(x)dx = \lim_{ \lambda \to 0} \sum _{i = 1}^{n}f(\xi) \Delta x_i \end{align}\)
二、性质：
<一>一般性质
\(\begin{align}&1. \int_a^b(f \pm g)dx = \int_a^b f dx \pm \int_a^b g dx \\ & 2. \int_a^b k f dx = k \int_a^b f dx \\ &3. \int_a^b f(x)dx = \int_a^cf(x)dx + \int_c^b f(x)dx \\ &4.\int_a^b 1 dx = b – a \\ &5.①f(x) \geq 0 (a < b) \Rightarrow \int_a^b f(x) dx \geq 0 \\ &②f \geq g (a < b) \Rightarrow \int_a^b f dx \geq \int_a^b g dx \\ &③|\int_a^b f dx| \leq \int_a^b |f| dx \\ &6.①f(x) \in c [a,b], \exists \xi \in [a,b] \Rightarrow \int_a^b f(x)dx = f(\xi)(b-a)(积分中值定理) \\&②f(x) \in c[a,b] ,\exists \xi \in (a,b) \Rightarrow \int_a^b f(x)dx = f(\xi)(b-a)\end{align}\)
<二>特殊性质
\(\begin{align}&1.f(x) \in [-a,a] \Rightarrow \int_{-a}^a f(x)dx = \int_0^a[f(x) + f(-x)]dx \\ &2.①\int_0^{\frac{\pi}{2}} f(\sin x)dx = \int_0^{\frac{\pi}{2}} f(\cos x)dx \\ &I_n = \int_0^{\frac{\pi}{2}} \cos ^n x dx = \int_0^{\frac{\pi}{2}} \sin ^n x dx \begin{cases} I_n = \frac{n-1}{n}I_{n-2} \\ I_0 = \frac{\pi}{2} \\ I_1 = 1\end{cases} \\ &②\int_0^{\pi} f(\sin x)dx = 2 \int_0^{\frac{\pi}{2}} f(\sin x)dx \\ &③\int_0^{\pi} x f(\sin x)dx = \frac{\pi}{2} \int_0^{\pi} f(\sin x)dx \\&3.f(x)以T为周期，则:\\&① \int_a^{a+T} f(x)dx = \int_0^T f(x)dx \\ &② \int_0^{nT} f(x)dx = n \int_0^T f(x)dx\end{align}\)
三、积分基本公式
Th1:\( f(x) \in c[a,b] , \Phi (x) = \int_a^x f(t)dt \Rightarrow \Phi^{\prime} (x)= f(x) \)
Th2: \( \int_a^bf(x)dx = F(b) – F(a)(牛顿-莱布尼茨公式)\)
四、广义积分：
1、区间无限（函数正常）情况下的反常积分
2、区间有限（函数在区间上有无穷间断点/瑕点）情况下的反常积分
