一元函数积分学
2022-05-16 16:04:22 0 举报AI智能生成
浙江省专升本数学-一元函数积分学
专升本高等数学题型
浙江省专升本
高等数学
模版推荐
作者其他创作
大纲/内容
不定积分
不定积分的基本概念与性质
原函数与不定积分的概念
原函数定义
如果在区间<span class="equation-text" contenteditable="false" data-index="0" data-equation="I"><span></span><span></span></span>上，可导函数<span class="equation-text" contenteditable="false" data-index="1" data-equation="F(x)"><span></span><span></span></span>的导函数为<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span>,即对任一<span class="equation-text" contenteditable="false" data-index="3" data-equation="x\in I"><span></span><span></span></span>,都有<span class="equation-text" contenteditable="false" data-index="4" data-equation="F^{'}(x)=f(x)"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="5" data-equation="dF(x)=f(x)dx"><span></span><span></span></span>，那么函数<span class="equation-text" contenteditable="false" data-index="6" data-equation="F(x)就称为f(x)(或f(x)dx)在区间I上的原函数"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x)在区间I内是连续的"><span></span><span></span></span>
由于<span class="equation-text" contenteditable="false" data-index="0" data-equation="[F(x)+C]^{'}=F^{'}(x)=f(x)，"><span></span><span></span></span>如果<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>有一个原函数，则有无穷多个原函数
一个函数的两个不同的原函数之间只差一个常数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)的全体原函数所组成的集合，就是函数族\{F(x)+C|-\propto<C<+\propto\}"><span></span><span></span></span>
不等积分的定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="在区间I上，函数f(x)的带有任意常数的原函数称为f(x)或f(x)dx在区间I上的不定积分，记作\int f(x)dx"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int 为积分号"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)为被积函数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)dx为被积表达式"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x为积分变量"><span></span><span></span></span>
原函数与不定积分的关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int f(x)dx=F(x)+C(其中C为任意常数，称为积分常数)"><span></span><span></span></span>
注
不定积分和原函数是两个不同的概念，不定积分是个集合，原函数是该集合中的一个元素，即<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int f(x)dx\neq F(x)"><span></span><span></span></span>
不定积分的运算性质
基本积分公式
不定积分的基本运算法则
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x)在区间I上有原函数存在，则kf(x)在区间I也有原函数存在，且\int kf(x)dx=k\int f(x)dx,其中k为任意常数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若函数f_1(x),f_2(x)在区间I上有原函数存在，则f_1(x)\pm f_2(x)在区间I也有原函数存在，且"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="\int [f_1(x)\pm f_2(x)]dx=\int f_1(x)dx\pm \int f_2(x)dx,可以推广到有限个和的形式"><span></span><span></span></span>
不定积分的计算方法
不定积分的换元积分法
第一类换元积分法（又称“凑微元”法）
定理1
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(u)有原函数，u=\phi(x)可导，则有换元公式"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int f[\phi(x)]\phi^{'}(x)dx=[f(u)du]_{u=\phi(x)}"><span></span><span></span></span>
注：常用的凑微元公式
第二类换元积分法（也称为变量代换法，t换元、三角换元）
定理2
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设x=\phi(t) 是单调的、可导的函数，并且\phi^{'}(t)\neq0,又设f[\phi(t)]\phi^{'}(t)具有原函数，则有换元公式\int f(x)dx=[f[\phi(t)]\phi^{'}(t)dt]]_{t=\phi^{-1}(x)}"><span></span><span></span></span>
第二换元法主要是针对包含有根式的被积函数，其左右是让根式有理化
一般采取以下几种代换
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}x=asint,t\in(-\frac{\pi}{2},\frac{\pi}{2}) \\x=atant,t\in (-\frac{\pi}{2},\frac{\pi}{2})\\x=asect ,t\in (0,\frac{\pi}{2})\end{cases}"><span></span><span></span></span>
注意：这种方式是利用了三角等式，称为三角代换，目的是将被积函数中的无理因式化为三角函数的有理因式
到代换<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=\frac{1}{t} ,利用它常可消去在被积函数f(x)的分母中的变量因子x^n"><span></span><span></span></span>
如果分母中有形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="x^{n}-a既要分大于a和小于a的情况再综上所述"><span></span><span></span></span>
当被积函数中含有无理式<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sqrt[n]{\frac{ax+b}{cx+d}}(a,b,c,d为实数)时，常代换t=\sqrt[n]{\frac{ax+b}{cx+d}}"><span></span><span></span></span>
不定积分的分布积分法
定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int udv=u\cdot v -\int vdu"><span></span><span></span></span>
常用分部积分法计算的不定积分类型
选<span class="equation-text" contenteditable="false" data-index="0" data-equation="u"><span></span><span></span></span>得规律
反对幂三指
改造分子法
解决形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int\frac{1}{sinx(x+a)sin(x+b)}dx,可以将分子为凑为sin[(x+a)-(x+b)]且乘常数\frac{1}{sin(a-b)}拆项"><span></span><span></span></span>
注
可以利用两角和两角差公式、当发现失衡了要及时换方法
简单有理函数积分
有理函数的积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设P_m(x)和Q_n(x)分别是m次和n次的实系数多项式，则形如式\frac{P_m(x)}{Q_n(x)}的函数称为有理函数。当n>m时，称为真分式，否则为假分"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="式"><span></span><span></span></span>
最简真分式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{A}{x-a}(a为常数)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{A}{(x-a)^{k}}(k>1,a为常数)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{Ax+B}{x^2+px+q}(p,q为常数，且p^2-4q<0)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{Ax+B}{(x^2+px+q)^{k}}(p,q为常数，且p^2-4q<0,k>1位整数)"><span></span><span></span></span>
注
显然最简真分式1、2恒容易求出，3、4需要凑一下如
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int \frac{x-2}{x^2+2x+3} \rightarrow \int \frac{\frac{1}{2}(x^2+2x+3)^{'}-3}{x^2+2x+3}dx"><span></span><span></span></span>
没有括号次方凑全
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int \frac{x+1}{(x^2+1)^2}dx \rightarrow \frac{1}{2}\int \frac{d(x^2+1)}{(x^2+1)^2}+\int \frac{dx}{(x^2+1)^2}"><span></span><span></span></span>
有括号次方凑括号里的
对于有理函数，通过以下程序求出原函数
如果式子式假分式，则可以表示成一个整式和一个真分式之和，然后分别求其原函数
如果式子为真分式，则可以将其分解成称为若干个最简分式之和，分别求出其原函数
关于将一个真分式分解成若干个最简分式之和，有如下定理
举个例子(重根例子)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{2x+2}{(x-1)(x^2+1)^2}=\frac{A}{x-1}+\frac{B_1x+C_1}{x^2+1}+\frac{B_2x+C_2}{(x^2+1)^2}"><span></span><span></span></span>
之后交叉相乘，通过比较系数的出系数，最后的出最简真分式
三角函数有理式的积分
定义
三角函数有理式是指三角函数和常数经过有限四则运算构成的函数。由于各种三角函数都可以用sinx以及cosx的有理式表示，故三角函数有理式也就是sinx，cosx的有理式，记作<span class="equation-text" contenteditable="false" data-index="0" data-equation="R(sinx,cosx)"><span></span><span></span></span>
对于三角函数有理函数的积分<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int R(sinx,cosx)dx,"><span></span><span></span></span>可通过万能代换化为有理函数的积分，具体方法：
<span class="equation-text" contenteditable="false" data-index="0" data-equation="取t=tan\frac{x}{2},则x=2arctant,dx=\frac{2}{1+t^2}dt."><span></span><span></span></span>
有三角函数中的万能公式，有
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx=\frac{2t}{1+t^2},cosx=\frac{1-t^2}{1+t^2}"><span></span><span></span></span>
因此有<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int R(sinx,cosx)dx=\int R(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2})\frac{2}{1+t^2}dt"><span></span><span></span></span>
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int R(x,\sqrt[n]{ax+b})dx(a\neq0)的积分[其中R(x,y)表示x,y的有理积分]"><span></span><span></span></span>，一般令<span class="equation-text" contenteditable="false" data-index="1" data-equation="t=\sqrt[n]{ax+b}，则可以化为有理积分"><span></span><span></span></span>
定积分
定积分的定义
定积分的概念
定积分的定义
考点：<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\rightarrow \propto} \sum_{i=1}^n f(\frac{i}{n})\cdot \frac{1}{n} = \int_{0}^1 f(x)dx"><span></span><span></span></span>
注
由定义知道<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^b f(x)dx"><span></span><span></span></span>是一个具体的数，与函数<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)"><span></span><span></span></span>以及区间<span class="equation-text" contenteditable="false" data-index="2" data-equation="[a,b]"><span></span><span></span></span>有关，而与积分变量x无关，即<span class="equation-text" contenteditable="false" data-index="3" data-equation="\int _a^b f(x)dx=\int _a^b f(u)du=\int _a^bf(t)dt"><span></span><span></span></span>
定义中的<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lambda\rightarrow 0 可以用n\rightarrow \propto代替"><span></span><span></span></span>
定积分的几何含义
定积分存在的充分条件与必要条件
充分条件
必要条件
定积分的性质
性质1
和差的定积分等于它的定积分的和差，即<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int a^b[f(x)\pm g(x)]dx=\int _a^bf(x)dx\pm \int _a^bg(x)dx"><span></span><span></span></span>
性质2
常数因子可以外提(可以推广到n个),<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^bkf(x)dx=k\int _a^bf(x)dx,k为任意常数"><span></span><span></span></span>
性质3
无论a,b,c的位置如何，有<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^bf(x)dx=\int _a^c f(x)dx + \int _c^b f(x)dx"><span></span><span></span></span>,即定积分对积分区间具有可加性，可以推广到有限个
性质4
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=1,则\int _a^bf(x)dx=b-a"><span></span><span></span></span>
性质5
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)\leq g(x)"><span></span><span></span></span>，则<span class="equation-text" contenteditable="false" data-index="1" data-equation="\int _a^b f(x)dx\leq \int _a^b g(x)dx,a<b"><span></span><span></span></span>
推论1.<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)\geq 0,x\in [a,b](a<b),则\int_a^bf(x)dx\geq0;"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="若在[a,b]上连续函数f(x)\geq0,且\int _a^bf(x)dx=0,则f(x)=0"><span></span><span></span></span>
性质6
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|\int _a^bf(x)dx| \leq \int _a^b |f(x)|dx(a<b)"><span></span><span></span></span>
性质7(积分估值定理)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设在[a,b],m\leq f(x)\leq M,则 m(b-a)\leq \int _a^bf(x)dx \leq M(b-a)"><span></span><span></span></span>
性质8(积分中值定理)
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在[a.b]上连续，则在[a,b]上至少存在一点\xi，使下式成立,\int _a^b f(x)dx=(b-a)f(\xi),a\leq \xi \leq b"><span></span><span></span></span>，也可以写成<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(\xi)=\frac{1}{b-a}\int _a^b f(x)dx，此式子称为f(x)在[a,b]的平均值公式"><span></span><span></span></span>
注
积分中值定理<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^b f(x)dx=(b-a)f(\xi)，a\leq \xi \leq b，不论a>b，还是a<b都是成立的"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="按积分中值定理公式可得f(\xi)=\frac{1}{b-a}\int _a^b f(x)dx，即函数f(x)在[a,b]的平均值公式"><span></span><span></span></span>
几何含义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="在区间[a,b]上至少存在一个\xi,使得以区间[a,b]为底边，以f(x)为曲边的曲边梯形的面积等于同一底边而高为f(\xi)的矩形面积"><span></span><span></span></span>
规定
由积分定义知，<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^b f(x)dx是当a<b时才有意义，而当a=b或a>b时无意义，规定"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a=b,\int _a^b f(x)dx=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a>b,\int _a^b=-\int _b^af(x)dx"><span></span><span></span></span>
定积分的计算方法
微积分基本公式
微积分基本定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设函数f(x)在[a,b]上连续，x\in [a,b],则函数f(x)在[a,x]上可积,以x为积分上限的定积分\int _a^x f(t)dt 称为变上限定积分"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x)在[-l,l]上是奇函数且连续,则\int _0^x f(t)dt 是偶函数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x)在[-l,l]上是偶函数且连续,则\int _0^x f(t)dt 是奇函数"><span></span><span></span></span>
定理3(微积分基本定理)
变上限定积分所确定的函数是被积函数的原函数,即<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在<span class="equation-text" contenteditable="false" data-index="1" data-equation="[a,b]"><span></span><span></span></span>上连续，<span class="equation-text" contenteditable="false" data-index="2" data-equation="x\in[a,b],则\phi(x)=\int _a^x f(t)dt 在[a,b]上可导，且导数为"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="3" data-equation="\phi^{'}(x)=\frac{d}{dx}(\int _a^xf(t)dt)=f(x) ,a\leq x \leq b"><span></span><span></span></span>
定理4
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果函数f(x)在[a,b]上连续，则积分上限的函数\phi(x)=\int _a^xf(t)dt 是f(x)在[a,b]上的一个原函数"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="变上限定积分的导数等于被积函数，这表明变上限定积分是被积函数的原函数"><span></span><span></span></span>
如果函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>在区间<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>上连续，那么在<span class="equation-text" contenteditable="false" data-index="2" data-equation="I"><span></span><span></span></span>上存在可导函数<span class="equation-text" contenteditable="false" data-index="3" data-equation="F(x)"><span></span><span></span></span>,使得<span class="equation-text" contenteditable="false" data-index="4" data-equation="F^{'}(x)=f(x),x\in I"><span></span><span></span></span>
某区间上的连续函数在该区间上存在原函数
既然变上限定积分是被积函数的原函数，这就为计算定积分开辟了新途径
牛顿-莱布尼兹公式（微积分基本公式）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果函数F(x)是连续函数f(x)在区间[a,b]上的一个原函数，则\int _a^b f(x)dx=F(b)-F(a)"><span></span><span></span></span>
注
牛顿-莱布尼兹公式把定积分的计算问题归结为被积函数的原函数在上、下限出函数值之差的问题
定积分的换元法和分部积分法
定积分的换元法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设函数f(x)在[a,b]上连续，函数x=\phi(t)满足条件"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi (\alpha) = a , \phi(\beta) = b"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi (x)在[\alpha,\beta](或[\beta,\alpha]) 上具有连续导数，且其值域R_\phi \subset [a,b]"><span></span><span></span></span>
则有<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^b f(x)dx = \int _{\alpha}^{\beta} f[\phi(t)] \phi^{'}(t)dt"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="用x=\phi(t)将原来积分变量为x的定积分化为积分变量为t的定积分时，积分上限也要换成新变量t的积分限"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求出f[\phi(t)]\phi^{'}(t)的一个原函数之后不必像计算不定积分时通过代回t=\phi^{-1}(x)变为原来变量x的函数，而是直接在新变量的原函数的基础上计算增量即可"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="换元公式也可以反过来用，即\int _a^bf[\phi^{-1}(x)][\phi^{-1}(x)]dx=\int _{\alpha}^{\beta} f(t)dt"><span></span><span></span></span>
从左往右看，是不定积分的第二换元法。从右往左看，可以认为是第一换元法
定积分的分部积分法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_a^b udv=[u\cdot v]_a^b -\int_a^b vdu"><span></span><span></span></span>
定积分计算中的常用公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x)在[-a,a]上连续，则\int _{-a}^{a} f(x)dx =\int _0^a[f(x)+f(-x)]dx = \begin{cases}0,f(x)为奇函数 \\2\int _0^af(x)dx,f(x)为偶函数 \end{cases}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x)在[0,1]上连续,则"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _0^{\frac{\pi}{2}} f(sinx)dx = \int _0^{\frac{\pi}{2}} f(cosx)dx"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _0^{\pi} xf(sinx)dx = \frac{\pi}{2} \int _0^{\pi} f(sinx)dx"><span></span><span></span></span>
区间再现公式
区间再现公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^bf(x)dx=\int _a^bf(a+b-x)dx=\frac{1}{2}\int _a^b[f(x)+f(a+b-x)]dx"><span></span><span></span></span>
证明的时候要么从第二式子出发令<span class="equation-text" contenteditable="false" data-index="0" data-equation="a+b-x=t,要么从第一个式子出发令x=a+b-t即可得证明"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="要么\int _a^bf(x)dx和\int _a^bf(a+b-x)dx都好算要么都很难算，\frac{1}{2}\int _a^b[f(x)+f(a+b-x)]dx会比\int _a^bf(x)dx和\int _a^bf(a+b-x)dx更好算"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=f(a+b-x)图像关于x=\frac{a+b}{2}对称，故再现公式第一个等号成立，第二个等号显然也是成立的，第二个等号表示为对称图形\frac{1}{2}显然是和前面等号成立"><span></span><span></span></span>
注
函数对称性
f(a-x)=f(x+b),则f(x)的图像关于x=(a+b)/2对称，<br>f(a-x)和f(x+b)的图像关于x=(a-b)/2对称。
当偶倍奇零、换元、凑微分、分部积分都无法使用，可以考虑再现公式
当遇到形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _0^1 arcsin2\sqrt{\frac{1}{4}-(x-\frac{1}{2})^2}dx的时候通过构造对称区间令x-\frac{1}{2}=t代换化简之后通过压缩区间令t=\frac{1}{2}cos\theta解决"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\rightarrow 0^+} xlnsinx=0，对数比x趋向0慢"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=sinx图形关于x=\frac{\pi}{2} 轴对称，\int _0^{\pi}sinxdx =2\int _0^{\frac{\pi}{2}}sinxdx,\int _0^{\pi}f(sinx)dx =2\int _0^{\frac{\pi}{2}}f(sinx)dx"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+sinx+cos)(1-sinx+cosx)=(1+cosx)^2-(sinx)^{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="对于P级数当p=2时，P级数收敛，且级数和1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots=\frac{\pi^2}{6}(P级数考求只有这个可以求)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="arcsin(sinx)=_{}^?x,\begin{cases}x\in(-\frac{\pi}{2},\frac{\pi}{2}),arcsin(sinx)=x\\x\in(\frac{\pi}{2},\pi),arcsin(sinx)=\pi-x\end{cases}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=arcsinx是y=sinx的反函数说法是错的，要分区间说。当区间不在"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="(-\frac{\pi}{2},\frac{\pi}{2})上时候，要拉回到这个区间上，可以利用诱导公式周期性拉回来"><span></span><span></span></span>
广义奇偶性：被积函数的对称中心横坐标恰好为区间中点
常函数(通过一阶导为0证明)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="arctanx+arctan\frac{1}{x}=\begin{cases}\frac{\pi}{2} ,x>0\\-\frac{\pi}{2},x<0\end{cases}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="形如arcsin\sqrt{1-x}+arcsin\sqrt{x}其中1-x+x=1是常函数，对其求导证明，代入特殊值点x=0得出常函数重新放到积分中去"><span></span><span></span></span>
黎曼积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="I=\int _{0}^{\frac{\pi}{2}}lnsinxdx=\int _{0}^{\frac{\pi}{2}}lncosdx=-\frac{\pi}{2}ln2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _0^{\frac{\pi}{2}} \frac{x}{tanx}dx=\frac{\pi}{2}ln2"><span></span><span></span></span>
经典题型1
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _{0}^{\frac{\pi}{2}}\frac{sinx}{sinx+cosx}dx(半角公式)、\int _{0}^{\frac{\pi}{2}}\frac{1}{1+(tanx)^\alpha}dx(tan(\frac{\pi}{2}-x)=cotx)、\int _{0}^{+\propto} \frac{1}{(1+x^2)(1+x^{\alpha})}dx(换元)"><span></span><span></span></span>
经典题型2
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _0^{n\pi} x|sinx|dx(正整数)利用区间再现得到I=\frac{n\pi}{2}\int _0^{n\pi} |sinx|dx=\frac{n\pi}{2}n\int _0^{\pi} sinxdx(利用图解法)"><span></span><span></span></span>
经典题型3
形如<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _0^1 (1-x)^{100}xdx(转移矛盾)通过使用区间再现公式转换出来"><span></span><span></span></span>
经典题型4
<span class="equation-text" contenteditable="false" data-index="0" data-equation="形如\int _0^2 x(x-1)(x-2)dx"><span></span><span></span></span>
解法一：区间再现
<span class="equation-text" contenteditable="false" data-index="0" data-equation="I=\int _0^2 x(x-1)(x-2)dx=\int _0^2 (2-x)(1-x)(-x)dx=-I\Rightarrow I=0"><span></span><span></span></span>
解法二：广义奇偶性
令<span class="equation-text" contenteditable="false" data-index="0" data-equation="x-1=t\Rightarrow \int _{-1}^{1}(1+t)t(t-1)dt=\int _{-1}^{1}t(t^2-1)dt=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x)是以T为周期的连续函数，则\int _{a}^{a+T} f(x)dx = \int _0^{T} f(x)dx ,\int _{a}^{a+nT} f(x)dx = n\int _0^{T} f(x)dx"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="I_n=\int _0^{\frac{\pi}{2}} sin^nxdx=\int _0^{\frac{\pi}{2}}cos^n xdx = \frac{n-1}{n}I_{n-2}\begin{cases}\frac{n-1}{n} \cdot\frac{n-3}{n-2} \cdots \frac{1}{2}\cdot \frac{\pi}{2},n为正偶数 \\\frac{n-1}{n} \cdot\frac{n-3}{n-2} \cdots \frac{4}{5}\cdot \frac{2}{3},n为大于1的正奇数 \end{cases}"><span></span><span></span></span>
无穷限广义积分
极限存在：收敛，极限不存在：发散
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若F^{'}(x)=f(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="则\int _a^{+\propto} f(x)dx= F(x)|_a^{+\propto} = \lim_{x\rightarrow +\propto} F(x) -F(a) "><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="则\int _{-\propto}^b f(x)dx= F(x)|_{-\propto}^{b} = F(b) -\lim_{x\rightarrow -\propto} F(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="则\int _{-\propto}^{+\propto} f(x)dx= F(x)|_{-\propto}^{+\propto} = \lim_{x\rightarrow +\propto} F(x) -\lim_{x\rightarrow -\propto}F(x) "><span></span><span></span></span>
瑕积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若c\in [a,b]且\lim_{x\rightarrow c}f(x) =\propto ,则x=c称为瑕点"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当a为唯一瑕点时"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^bf(x)dx=F(x)|_{a^{+}}^b=F(b)-\lim_{x\rightarrow a^{+}}F(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当b为唯一瑕点时"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^bf(x)dx=F(x)|_{a^{}}^b{^{-}}=\lim_{x\rightarrow b^{-}}F(b)-F(a)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当瑕点只有a和b时"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^bf(x)dx=F(x)|_{a^{+}}^{b^{-}}=\lim_{x\rightarrow b^{-}}F(b)-\lim_{x\rightarrow a^{+}}F(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="当c(c<c<b)为唯一瑕点时"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int _a^bf(x)dx=\int _a^cf(x)dx+\int _c^bf(x)dx=F(x)|_a^{c^{-}}+F(x)|_{c^{+}}^{b}=\lim_{x\rightarrow c^{-}}F(x)-F(a)+F(b)-\lim_{x\rightarrow c^{+}}F(x)"><span></span><span></span></span>
定积分的应用
定积分应用的微元法
平面图形的面积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="区域D由连续曲线y=f(x),y=g(x),和直线x=a,x=b围成，其中f(x)\leq g(x)(a\leq x \leq b),D的面积为A=\int _a^b [f(x)-g(x)]dx=\int _a^b [上-下]dx"><span></span><span></span></span>
旋转体的体积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="绕x轴"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="V_x=\pi\int _a^bf^2(x)dx "><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="绕y轴"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="V_y=\pi\int _c^dg^2(y)dy"><span></span><span></span></span>
柱壳法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="V_y=2\pi\int_a^b x\cdot|f(x)|dx"><span></span><span></span></span>
特殊的情况
立体图形体积
圆柱体积
V=π*r²* h
圆锥体积
V=1/3Sh
圆台体积公式
V=1/3πh（r²＋R²＋rR）
Collect
Get Started
Collect
Get Started
Collect
Get Started
Collect
Get Started
评论
0 条评论
下一页
