CAL-5-定积分
2021-08-01 14:39:44 0 举报AI智能生成
高等数学微积分 第五章 定积分 知识点梳理
作者其他创作
大纲/内容
积分法
换元法
令a=φ(α)，b=φ(β)
<br><span class="equation-text" data-index="0" data-equation="\int_a^bf(x)\mathrm dx=\int_{\alpha}^{\beta}f[\varphi(x)]\varphi '(x)\mathrm dx" contenteditable="false"><span></span><span></span></span>
分部积分法
<span class="equation-text" data-index="0" data-equation="\int_{a}^{b}u\mathrm{d}v=uv\big|_a^b-\int_a^bv\mathrm d u" contenteditable="false"><span></span><span></span></span>
特殊性质
奇偶性(1)
<span class="equation-text" data-index="0" data-equation="\int _{-a}^a f(x)\mathrm dx=\int _{0}^a [f(x)+f(-x)]\mathrm dx" contenteditable="false"><span></span><span></span></span>
有理化:<span class="equation-text" data-index="0" data-equation="\int_{-1}^1even(x)\ln(x+\sqrt{1+x^2})\mathrm dx" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="=\int_0^1even(x)\ln(x+\sqrt{1+x^2})\mathrm dx+\int_{-1}^0even(x)\ln(x+\sqrt{1+x^2})\mathrm dx\\" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="=\int_0^1even(x)\ln1\mathrm dx=0" contenteditable="false"><span></span><span></span></span>
三角函数(4)
设f(x)∈C[0,1]，满足<span class="equation-text" data-index="0" data-equation="\int _0 ^\frac{\pi}{2} f(\sin x)\mathrm dx=\int _0 ^\frac{\pi}{2} f(\cos x)\mathrm dx" contenteditable="false"><span></span><span></span></span>
通常遇到分母变换后仍不变的，可以考虑此变换<br>
<br><span class="equation-text" data-index="0" data-equation="记\it I _n=\int_0^\frac{\pi}{2} sin^n x\mathrm dx" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\it I_n=\frac{n-1}{n}\it I_{n-2}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\it I_0=\frac{\pi}{2}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\it I_1 =1" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int _0^\pi f(\sin x)\mathrm dx = 2\int _0^{\frac \pi 2} f(\sin x)\mathrm dx" contenteditable="false"><span></span><span></span></span>
对f(cosx)不适用
<span class="equation-text" data-index="0" data-equation=" \int _0^\pi x\sin x\mathrm dx=\frac{\pi}2 \int _0^\pi \sin x\mathrm dx" contenteditable="false"><span></span><span></span></span>
可用区间再现公式证明
<span class="equation-text" data-index="0" data-equation="出现 \sin x \cos x \mathrm dx一般考虑变成\sin x\mathrm d \sin x" contenteditable="false"><span></span><span></span></span>
周期性(2)
<span class="equation-text" data-index="0" data-equation="\int_a^{a+T}f(x)\mathrm dx=\int_0^Tf(x)\mathrm dx" contenteditable="false"><span></span><span></span></span>：平移不变性
<span class="equation-text" data-index="0" data-equation="\int_0^{nT}f(x)\mathrm dx=n\int_0^Tf(x)\mathrm dx" contenteditable="false"><span></span><span></span></span>：倍周期
区间再现公式
<span class="equation-text" data-index="0" data-equation="\int _a^b f(x)\mathrm dx =\int _a^b f(a+b-x)\mathrm dx" contenteditable="false"><span></span><span></span></span>
易错点
不要把dx的x也换了
原理和证明
令t=a+b-x，最后把t换成x
反常积分
无限区间
[a,+∞)
(-∞,b]
(-∞,+∞)
判别法<br>Φ(x)=x^α * f(x)
α>1，且limΦ(x)存在→收敛
α≤1，且limΦ(x)=k(≠0)或∞→发散
第二类间断点(无穷间断点)
f(x)∈C(a,b]，f(a+0)=∞
判别法<br><span class="equation-text" data-index="0" data-equation="\Phi(x)=(x-a)^αf(x)" contenteditable="false"><span></span><span></span></span>
α<font color="#c41230"><</font>1，且<span class="equation-text" data-index="0" data-equation="\lim\limits_{x\rightarrow a^+}\Phi(x) " contenteditable="false"><span></span><span></span></span>存在→收敛
α≥1，且<span class="equation-text" data-index="0" data-equation="\lim\limits_{x\rightarrow a^+}\Phi(x) " contenteditable="false"><span></span><span></span></span>=k(>0)或∞→发散
f(x)∈C[a,b)，f(b-0)=∞
判别法<br><span class="equation-text" data-index="0" data-equation="\Phi(x)=(b-x)^αf(x)" contenteditable="false"><span></span><span></span></span>
α<font color="#c41230"><</font>1，且<span class="equation-text" data-index="0" data-equation="\lim\limits_{x\rightarrow b^-}\Phi(x) " contenteditable="false"><span></span><span></span></span>存在→收敛
α≥1，且<span class="equation-text" data-index="0" data-equation="\lim\limits_{x\rightarrow b^-}\Phi(x) " contenteditable="false"><span></span><span></span></span>=k(>0)或∞→发散
Γ函数
<span class="equation-text" data-index="0" data-equation="\Gamma(\alpha)=\int_0^{+\infin}x^{\alpha-1}e^{-x}\mathrm dx" contenteditable="false"><span></span><span></span></span>
性质
<span class="equation-text" data-index="0" data-equation="\Gamma(\alpha+1)=\alpha\Gamma(\alpha)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\Gamma(n+1)=n!" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\Gamma(1/2)=\sqrt{\pi}" contenteditable="false"><span></span><span></span></span>
要点<br>
核心是积分定义极限存在，因此对于既有无限区间又有间断点的积分，需要论证无穷远和间断点的极限均存在
两个α构造的Φ(x)可能是在另一方向上指向相反结论的，只需要保证两个Φ(x)在任意的分别不同方向上收敛就可以了。
同时存在两个端点是第二类间断点时，α可能是同一个值。
基本定义
f(x)在[a,b]上有界
f(x)在[a,b]上有界是可积的必要不充分条件
<span class="equation-text" data-index="0" data-equation="\int_a^bf(x)\mathrm dx=\lim\limits_{\Delta x_i\rightarrow0}\sum\limits_{i=1}^nf(\xi_i)\Delta x_i" contenteditable="false"><span></span><span></span></span>
极限存在→可积<br>
极限不存在→不可积
基本形态(3)
<span class="equation-text" data-index="0" data-equation="\int_0^1f(x)\mathrm dx=\lim\limits_{n\rightarrow\infin}\frac 1 n \sum\limits_{i=0} ^nf(\frac i n) =\lim\limits_{n\rightarrow\infin}\frac 1 n \sum\limits_{i=1} ^nf(\frac {i-1} n) " contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="\int_0^kf(x)\mathrm dx=\lim\limits_{n\rightarrow\infin}\frac 1 n\sum\limits _{i=0}^{kn} f(\frac i n) " contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^bf(x)\mathrm dx=\lim\limits_{n\rightarrow\infin}\frac {b-a}n\sum\limits _{i=0}^n f(a+\frac i n(b-a)) " contenteditable="false"><span></span><span></span></span>
积分基本定理
牛顿-莱布尼茨公式
<span class="equation-text" data-index="0" data-equation="\int_a^bf(x)\mathrm dx=F(x)\big|_a^b" contenteditable="false"><span></span><span></span></span>
设f(x)∈C[a,b]，<span class="equation-text" data-index="0" data-equation="\Phi(x)=\int_a^xf(t)\mathrm dt" contenteditable="false"><span></span><span></span></span>，则<span class="equation-text" data-index="1" data-equation="\frac{\mathrm d}{\mathrm dx}\Phi(x)=f(x)" contenteditable="false"><span></span><span></span></span>，ie. <span class="equation-text" data-index="2" data-equation="\Phi'(x)=f(x)" contenteditable="false"><span></span><span></span></span><br>变积分限函数/积分变限函数<br>
变积分限函数
<span class="equation-text" data-index="0" data-equation="\frac {\mathrm d}{{\mathrm dx}}\int_a^{x} f(t){\mathrm dt}=f(x)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\frac {\mathrm d}{{\mathrm dx}}\int_a^{\varphi (x)} f(t){\mathrm dt}=f[\varphi(x)]\varphi'(x)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\frac {\mathrm d}{{\mathrm dx}}\int_{\varphi_1(x)}^{\varphi_2(x)} f(t){\mathrm dt}=f[\varphi_2(x)]\varphi'_2(x)-f[\varphi_1(x)]\varphi'_1(x)" contenteditable="false"><span></span><span></span></span>
积分中值定理
设f(x)∈C[a,b]，则存在ξ∈(a,b)使<span class="equation-text" data-index="0" data-equation="\int_a^bf(x)\mathrm dx=f(\xi)(b-a)" contenteditable="false"><span></span><span></span></span>
基本初等函数的定积分
<span class="equation-text" data-index="0" data-equation="\int_a^b k \mathrm dx=kx|_a^b" contenteditable="false"><span></span><span></span></span>
幂，指(4)<br>
<span class="equation-text" data-index="0" data-equation="\int_a^b x\mathrm dx=\frac 1 2x^2|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b x^a\mathrm dx=\frac {x^{a+1}} {a+1}|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b e^x \mathrm dx=e^x|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b a^x \mathrm dx=\frac{a^x}{\ln{a}}|_a^b" contenteditable="false"><span></span><span></span></span>
三角函数(10)<br>
<span class="equation-text" data-index="0" data-equation="\int_a^b\sin x\mathrm dx=-\cos x|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\cos x\mathrm dx=\sin x|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\tan x\mathrm dx=-\ln{|\cos x|}|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\cot x\mathrm dx=\ln{|\sin x|}|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\sec x\mathrm dx=\ln|\sec x+\tan x||_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\csc x\mathrm dx=\ln|\csc x-\cot x||_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\sec^2x\mathrm dx=\tan x|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\csc^2x\mathrm dx=-\cot x|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\sec x\tan x\mathrm dx=\sec x|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\csc x\cot x\mathrm dx=-\csc x|_a^b" contenteditable="false"><span></span><span></span></span>
平方和/差(2)<br>根号平方和/差(5)<br>
<span class="equation-text" data-index="0" data-equation="\int_a^b\frac{\mathrm dx}{x^2+a^2}=\frac 1 a\arctan{\frac x a}\big|_a^b(a\neq0)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\frac{\mathrm dx}{x^2-a^2}=\frac 1 {2a}\ln|\frac {x-a}{x+ a}|\big|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\frac{\mathrm dx}{\sqrt{a^2-x^2}}=\arcsin{\frac x a}\big|_a^b(a>0)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b{\sqrt{a^2-x^2}}\mathrm dx=\frac x 2\sqrt{a^2-x^2}\big|_a^b+\frac{a^2}2\arcsin\frac x a\big|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\frac{\mathrm dx}{\sqrt{x^2+a^2}}=\ln(x+\sqrt{x^2+a^2})\big|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b\frac{\mathrm dx}{\sqrt{x^2-a^2}}=\ln(x+\sqrt{x^2-a^2})\big|_a^b" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int_a^b{\sqrt{a^2+x^2}}\mathrm dx=\frac x 2\sqrt{a^2+x^2}\big|_a^b+\frac{a^2}2\ln|x+\sqrt{a^2+x^2}|\big|_a^b" contenteditable="false"><span></span><span></span></span>
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