一、定义：

\begin{array}{r} \int_{a}^{b} f (x) d x = lim_{λ \to 0} \sum_{i = 1}^{n} f (ξ) Δ x_{i} \end{array}

二、性质：

\begin{aligned} 1. \int_{a}^{b} (f \pm g) d x = \int_{a}^{b} f d x \pm \int_{a}^{b} g d x \\ 2. \int_{a}^{b} k f d x = k \int_{a}^{b} f d x \\ 3. \int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x \\ 4. \int_{a}^{b} 1 d x = b - a \\ 5. ① f (x) \geq 0 (a < b) \Rightarrow \int_{a}^{b} f (x) d x \geq 0 \\ ② f \geq g (a < b) \Rightarrow \int_{a}^{b} f d x \geq \int_{a}^{b} g d x \\ ③ | \int_{a}^{b} f d x | \leq \int_{a}^{b} | f | d x \\ 6. ① f (x) \in c [a, b], \exists ξ \in [a, b] \Rightarrow \int_{a}^{b} f (x) d x = f (ξ) (b - a) (积 分 中 值 定 理) \\ ② f (x) \in c [a, b], \exists ξ \in (a, b) \Rightarrow \int_{a}^{b} f (x) d x = f (ξ) (b - a) \end{aligned}

\begin{aligned} 1. f (x) \in [- a, a] \Rightarrow \int_{- a}^{a} f (x) d x = \int_{0}^{a} [f (x) + f (- x)] d x \\ 2. ① \int_{0}^{\frac{π}{2}} f (\sin x) d x = \int_{0}^{\frac{π}{2}} f (\cos x) d x \\ I_{n} = \int_{0}^{\frac{π}{2}} \cos^{n} x d x = \int_{0}^{\frac{π}{2}} \sin^{n} x d x {\begin{cases} I_{n} = \frac{n - 1}{n} I_{n - 2} \\ I_{0} = \frac{π}{2} \\ I_{1} = 1 \end{cases} \\ ② \int_{0}^{π} f (\sin x) d x = 2 \int_{0}^{\frac{π}{2}} f (\sin x) d x \\ ③ \int_{0}^{π} x f (\sin x) d x = \frac{π}{2} \int_{0}^{π} f (\sin x) d x \\ 3. f (x) 以 T 为 周 期 ， 则 : \\ ① \int_{a}^{a + T} f (x) d x = \int_{0}^{T} f (x) d x \\ ② \int_{0}^{n T} f (x) d x = n \int_{0}^{T} f (x) d x \end{aligned}