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一元函数积分学 - Coggle Diagram
一元函数积分学
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定积分的定义与性质
定积分的性质
定积分的基本性质
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3.积分的不等式性
推论1：\((g(x)=0\)的特殊情况)
若函数\(f(x)\)在[a,b]上可积，且\(f(x) = 0\)，则\(\int_a^b {f(x)dx} \le 0 \)
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若函数\(f(x)\),\(g(x)\)在[a,b]上可积，且\(f(x) \le g(x)\)，则\(\int_a^b {f(x)dx} \le \int_a^b {g(x)dx} \)
推论3 :!:
若函数\(f(x)\),\(g(x)\)在[a,b]上可积，且\(|f(x)|\)也可积，
则|\(\int_a^b {f(x)dx} |\le \int_a^b {|g(x)|dx} \)
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定积分基本性质的推广
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2.积分第一中值定理\(f(x),g(x)\) #
3:!:.设若函数\(f(x)\)在[a,b]上可连续，\(f(x)\)非负且\(\int_a^b {f(x)dx} =0\)，则\(f(x) \equiv 0\).
4:!:.设若函数\(f(x)\)在[a,b]上可连续，\(f(x)\)不恒为0，则\(\int_a^b {f(x)dx} >0\).
三角函数定积分性质
设若函数\(f(x)\)在[0,1]上可连续,
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2. :!:
\(\int_0^{\frac{\pi }{2}} {f(\sin x)dx} = \int_0^{\frac{\pi }{2}} {f(\cos x)dx}=\int_{\frac{\pi }{2}}^\pi {f(\sin x)dx}\)
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3. :!:
\(\int_0^\pi {f(\sin x)dx} = 2\int_0^{\frac{\pi }{2}} {f(\sin x)dx}\)
\(\int_0^{2\pi } {f(|\sin x|)dx} = 4\int_0^{\frac{\pi }{2}} {f(\sin x)dx}\)
4. :!:
\(\int_0^\pi {f(|cos x|)dx} = 2\int_0^{\frac{\pi }{2}} {f(\cos x)dx}\)
\(\int_0^{2\pi } {f(|\cos x|)dx} = 4\int_0^{\frac{\pi }{2}} {f(\cos x)dx}\)
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定积分的定义
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定积分定义法求极限
\(\mathop {\lim }\limits_{n \to \infty } \sum\limits_{{\rm{i}} = 0}^n {f(a + \frac{{i(b - a)}}{n})\frac{{b - a}}{n} = \int_a^b {f(x)dx} } \)
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