《高等数学》读书笔记
2021-08-24 11:49:17 74 举报AI智能生成
高等数学极限、积分等知识点总结
作者其他创作
大纲/内容
杂项
约定当n趋向∞时，默认为趋向+∞
常用不等式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sin x<x<\tan x,x\in(0,\pi/2)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{x}{1+x}<\ln(1+x)<x,x>0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^x\ge1+x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="2ab\le a^2+b^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sqrt[3]{abc}\le\frac{1}{3}(a+b+c)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="|a+b|\le|a|+|b|"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1^2+2^2+\cdots+n^2=\frac{1}{6}n(n+1)(2n+1)"><span></span><span></span></span>
函数 极限 连续
函数
概念及常见函数
反函数
函数性态判定
单调性
利用定义
利用导数<br>设f(x)在区间I可导，则<br><span class="equation-text" data-index="0" data-equation="1.f^\prime(x)>0(<0)\Rightarrow f(x)单调增(单调减);" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="2.f^\prime(x)\geq0(\leq0)\Leftrightarrow f(x)单调不减(单调不增);"><span></span><span></span></span><br>
奇偶性
利用定义
若f(x)可导，且f(x)是奇(偶)函数，则求导后导函数奇偶性改变
连续的奇函数，其原函数都是偶函数；<br>连续的偶函数，其原函数中有唯一一个是奇函数（因为加个常数C就不是奇函数了）<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1)若f(x)是奇函数，则\begin{matrix} \int_{_0}^{x} f(t)\, \mathrm{d}t\end{matrix}是偶函数"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="(2)若f(x)是偶函数，则\begin{matrix} \int_{_0}^{x} f(t)\, \mathrm{d}t\end{matrix}是奇函数"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="若f(x)是奇函数，则\begin{matrix} \int_{_a}^{x} f(t)\, \mathrm{d}t\end{matrix}也是偶函数" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="因为\begin{matrix} \int_{_a}^{x} f(t)\, \mathrm{d}t\end{matrix}=\begin{matrix} \int_{_a}^{0} f(t)\, \mathrm{d}t\end{matrix}(常数)+\begin{matrix} \int_{_0}^{x} f(t)\, \mathrm{d}t\end{matrix}(偶函数)"><span></span><span></span></span><br>
奇函数若在x=0有定义，则f(0)一定等于0
周期性
用定义f(x+T)=f(x)
可导的周期函数其导函数为周期函数
周期函数的原函数是周期函数的充要条件是其在一个周期上的积分为0
有界性
用定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在[a,b]上连续\Rightarrow f(x)在[a,b]上有界"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)在(a,b)上连续，且左端点右极限f(a^+)和右端点左极限f(b^-)存在\Rightarrow f(x)在(a,b)上有界"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="注：区间(a,b)改为无穷区间(-∞,b),(a,+∞),(-∞,+∞)结论仍成立"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f^\prime(x)在区间I(有限区间，即端点不为∞)上有界\Rightarrow f(x)在I上有界"><span></span><span></span></span>
极限
极限概念
数列极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n \to \infty}x_n=a\Leftrightarrow \lim_{k\to \infty}x_{2k-1}=\lim_{k \to \infty}x_{2k}=a"><span></span><span></span></span>
函数极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x \to \infty}f(x)=A\Leftrightarrow\lim_{x \to +\infty}f(x)=\lim_{x \to -\infty}f(x)=A"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x \to x_{_0}}f(x)=A\Leftrightarrow\lim_{x \to x_{_0}^+}f(x)=\lim_{x \to x_{_0}^-}f(x)=A"><span></span><span></span></span>
三种需要分左右极限的情况
分段函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^∞"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\arctan∞"><span></span><span></span></span>
极限性质
局部有界性
保号性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设\lim_{x \to x_{_0}}f(x)=A，则"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="若A>0\Rightarrow x_{_0}的某去心邻域内f(x)>0"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="2" data-equation="若x_{_0}的某去心邻域内f(x)\geq0\Rightarrow A\geq0"><span></span><span></span></span>
极限值与无穷小之间的关系
极限存在准则
夹逼准则
单调有界准则
无穷小
无穷大
常<对<幂<指<阶
求极限常用方法
有理运算法则
基本极限
等价无穷小
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若lim\frac{\alpha_1}{\beta_1}\not=1，则\alpha-\beta\sim\alpha_1-\beta_1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若lim\frac{\alpha_1}{\beta_1}\not=-1，则\alpha+\beta\sim\alpha_1+\beta_1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x)和g(x)在x=0的某邻域内连续，且lim\frac{f(x)}{g(x)}=1，则"><span></span><span></span></span><br> <span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{matrix} \int_{_0}^{x} f(x)\, \mathrm{d}x \end{matrix}\sim\begin{matrix} \int_{_0}^{x} g(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
洛必达
泰勒
夹逼
定积分定义
单调有界准则
中值定理
微分中值定理
拉格朗日
积分中值定理
函数极限
0/0
∞/∞
∞-∞
提取无穷因子
有理化
化为0/0
0·∞
可以先用等价无穷小处理0
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1^∞"><span></span><span></span></span>
三步曲
<span class="equation-text" contenteditable="false" data-index="0" data-equation="∞^0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0^0"><span></span><span></span></span>
数列极限
不定式
改写成函数极限
n项和的数列极限
夹逼定理
定积分定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_0}^{1} f(x)\, \mathrm{d}x \end{matrix}=\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^n f(\frac{i}{n})"><span></span><span></span></span><br>积分区间通过原式判断
级数求和
n项连乘的数列极限
夹逼定理
取对数化为n项和
递推关系<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1=a,x_{n+1}=f(x_n)(n=1,2,\cdots)"><span></span><span></span></span>定义的数列
法1:先用单调有界准则证明极限存在，再在递推关系两端取极限
法2:先令极限为A，根据递推关系解得A后证明极限为A
证明思路：<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="核心是利用x_n=f(x_{n-1})证明一个递推不等式"><span></span><span></span></span><br> <span class="equation-text" contenteditable="false" data-index="1" data-equation="|x_n-a|\le A|x_{n-1}-a|（0<A<1）"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="2" data-equation="最终A^{n-1}|x_1-a|\to 0（n\rightarrow∞）则得证"><span></span><span></span></span>
单调性判定<br>
后项-前项
后项/前项
1.若f(x)单增，则<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="当x_1\le x_2,{x_n}单增，当x_1\ge x_2,{x_n}单减"><span></span><span></span></span><br>2.若f(x)单减，则该数列单调性不存在
若已知递推关系f(x)导数大于0，则数列一定单调，此时也可证数列即有上界又有下界即可
证明函数有界
常见不等式
归纳法
处理根号-根号
有理化
拉格朗日中值定理
提后面的根号，利用等价无穷小<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+f(x))^\alpha-1\sim\alpha f(x) （f(x)趋向0）"><span></span><span></span></span>
连续
间断点
第一类间断点（极限存在）
可去间断点：左右极限存在且相等
跳跃间断点：左右极限存在但不相等
第二类间断点（极限不存在）
无穷间断点：左右极限有一个无穷
振荡间断点：如x=0为f(x)=sin1/x的振荡间断点
性质
有界
最值
介值定理
f(x)可取到区间上最大最小值之间的任一一值
零点定理
1.f(x)在区间上连续2.且在区间两端点异号，则必存在零点
一元函数微分学
导数与微分
题目出现f(x)n阶可导，则洛必达最多能使用到出现n-1阶<br>题目出现f(x)n阶连续可导，则洛必达最多能使用到出现n阶<br>若还是无法解出，则一般用导数定义
基本概念
导数：某点变化率<br>几何意义：某点切线斜率
微分：函数改变量近似值
题型一 导数与微分概念
导数定义求极限
导数定义求导数
（难点）利用导数定义判定可导性
<span class="equation-text" data-index="0" data-equation="由\lim_{x\to x_0}\frac{f(\alpha(x))-f(x_0)}{\beta(x)}存在\Rightarrow f(x)在x_0可导，需满足以下条件" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="1" data-equation="1.\alpha(x)\to 0,\alpha(x)\not=0" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="2" data-equation="2.\alpha(x)能从x_0两侧趋向" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="3" data-equation="3.\alpha(x)与\beta(x)同阶，因为上式子乘除一个\alpha(x)，会形成导数定义及\frac{\alpha(x)}{\beta(x)}"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="设f(x)=\beta(x)｜x-a｜,\beta(x)在a点连续，则" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x)在x=a处可导的充要条件是\beta(a)=0"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="由|x|在x=0处不可导，x|x|在x=0处可导，知" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="(x-a)|x-a|在x=a处可导"><span></span><span></span></span>
题型二 导数的几何意义
题型三导数与微分的计算
复合、隐、反
参数方程
当题目为隐函数与参数方程求导的综合题时<br>往往求二阶导数代公式比较简单
对数求导法
适用于幂指函数，连乘，连除，开方，乘方等形式
高阶导数
代公式p51
求1、2阶导数，归纳法
利用泰勒级数或泰勒公式
导数应用
基础概念
罗尔
拉格朗日
柯西
泰勒
极值
驻点及导数不存在的点可能取到极值
最值
一元函数积分学
不定积分
三种主要积分法
第一换元积分
第二换元积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sqrt{(a^2-x^2)^\alpha}"><span></span><span></span></span>
令<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=a\sin t"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\sqrt{(a^2+x^2)^\alpha}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=a\tan t"><span></span><span></span></span>
也可消去分母中(<span class="equation-text" data-index="0" data-equation="a^2+(x+b)^2" contenteditable="false"><span></span><span></span></span>)的高次幂<br>例：<span class="equation-text" data-index="1" data-equation="\begin{matrix} \int_{}^{} \frac{x^3}{_{(x^2-2x+2)^2}}\, \mathrm{d}x \end{matrix}" contenteditable="false"><span></span><span></span></span><br>其中<br><span class="equation-text" data-index="2" data-equation="x^2-2x+2=(x-1)^2+1" contenteditable="false"><span></span><span></span></span><br>令<span class="equation-text" data-index="3" data-equation="x-1=\tan t" contenteditable="false"><span></span><span></span></span>即可
<span class="equation-text" data-index="0" data-equation="\sqrt{(x^2-a^2)^\alpha}" contenteditable="false"><span></span><span></span></span>
令<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=a\sec t"><span></span><span></span></span>
分部积分法
凑多项式以外的
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} P_n(x)e^{\alpha x}\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} P_n(x)\sin \alpha x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} P_n(x)\cos \alpha x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
凑多项式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} P_n(x)\ln x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} P_n(x)\arctan x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} P_n(x)\arcsin x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
凑指数或三角皆可
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} e^{\alpha x}\sin \beta x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{}^{} e^{\alpha x}\cos \beta x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
三类常见可积积分
有理函数积分<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{}^{} R(x)\, \mathrm{d}x"><span></span><span></span></span>
一般方法(部分分式法)
对于真分式<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{P(x)}{Q(x)}"><span></span><span></span></span><br>若分母能分解为两个多项式乘积<span class="equation-text" contenteditable="false" data-index="1" data-equation="Q_1(x)Q_2(x)"><span></span><span></span></span><br>则能将该真分式拆成两个真分式之和，然后再分别积分
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q_1(x)=ax+b"><span></span><span></span></span><br>则拆分后为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{A}{ax+b}"><span></span><span></span></span>
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q_1(x)=ax^2+bx+c"><span></span><span></span></span><br>则拆分后为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{Ax+B}{ax^2+bx+c}"><span></span><span></span></span>
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q_1(x)=ax^4+..."><span></span><span></span></span><br>则拆分后为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{Ax^2+Bx+C}{ax^4+...}"><span></span><span></span></span>
特殊方法
凑微分降幂
加项减项拆
三角有理式积分<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{}^{} R(\sin x,\cos x)\, \mathrm{d}x"><span></span><span></span></span>
一般方法(万能变换)
令<span class="equation-text" data-index="0" data-equation="\tan \frac{x}{2}=t" contenteditable="false"><span></span><span></span></span><br>则<span class="equation-text" data-index="1" data-equation="\sin x=\frac{2t}{1+t^2}" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="2" data-equation="\cos x=\frac{1-t^2}{1+t^2}" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="3" data-equation="dx=\frac{2}{1+t^2}dt"><span></span><span></span></span><br>
特殊方法
三角变形
换元
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="R(-\sin x,\cos x)=-R(\sin x,\cos x)"><span></span><span></span></span>,则令<span class="equation-text" contenteditable="false" data-index="1" data-equation="u=\cos x"><span></span><span></span></span>,或凑<span class="equation-text" contenteditable="false" data-index="2" data-equation="d\cos x"><span></span><span></span></span>
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="R(\sin x,-\cos x)=-R(\sin x,\cos x)"><span></span><span></span></span>,则令<span class="equation-text" contenteditable="false" data-index="1" data-equation="u=\sin x"><span></span><span></span></span>,或凑<span class="equation-text" contenteditable="false" data-index="2" data-equation="d\sin x"><span></span><span></span></span>
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="R(-\sin x,-\cos x)=R(\sin x,\cos x)"><span></span><span></span></span>,则令<span class="equation-text" contenteditable="false" data-index="1" data-equation="u=\tan x"><span></span><span></span></span>,或凑<span class="equation-text" contenteditable="false" data-index="2" data-equation="d\tan x"><span></span><span></span></span>
分部积分
简单无理函数积分<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{}^{} R(x,\sqrt[n]{\frac{ax+b}{cx+d}})\, \mathrm{d}x"><span></span><span></span></span>
令<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sqrt[n]{\frac{ax+b}{cx+d}}=t"><span></span><span></span></span>
常见“积不出”
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{}^{} e^{x^2}\, \mathrm{d}x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{}^{} \frac{\sin x}{x}\, \mathrm{d}x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{}^{} \frac{\cos x}{x^n}\, \mathrm{d}x"><span></span><span></span></span>
其他
得到答案<span class="equation-text" contenteditable="false" data-index="0" data-equation="\ln \frac{\sqrt{\alpha(x)}-1}{\sqrt{\alpha(x)}+1}"><span></span><span></span></span>可以不用继续化简
得到的答案中有多个<span class="equation-text" contenteditable="false" data-index="0" data-equation="\ln \alpha(x)"><span></span><span></span></span>，应整合到一起
原函数存在定理：<br>若f(x)在区间i上连续，则f(x)在区间i上一定存在原函数<br>若f(x)在区间i上有第一类间断点，则f(x)在区间i上没有原函数<br>
由原函数定义<span class="equation-text" contenteditable="false" data-index="0" data-equation="F^\prime(x)=f(x)"><span></span><span></span></span>可知，若f(x)在(a,b)上有原函数F(x)，则F(x)在(a,b)一定连续。
求分段函数的原函数时，需要注意加减常数，使得原函数在分段点连续
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\cos x=\cos^2\frac{x}{2}-\sin^2\frac{x}{2}"><span></span><span></span></span>
定积分
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>几何意义
值为f(x)曲线与x=a,x=b及x轴围成的区域，在x轴上方的面积减去下方的面积所得之差
性质
不等式性质
若在区间[a,b]上<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="\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}\leq \begin{matrix} \int_{_a}^{b} g(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
若M和m是f(x)在区间[a,b]上的最小值，则<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="m(b-a)\leq\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}\leq M(b-a)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\left|\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}\right|\leq \begin{matrix} \int_{_a}^{b} \left|f(x)\right|\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
积分中值定理
若f(x)在[a,b]上连续，则<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}=f(\xi)(b-a)"><span></span><span></span></span> <span class="equation-text" contenteditable="false" data-index="1" data-equation="(a<\xi<b)"><span></span><span></span></span>
若f(x),g(x)在[a,b]上连续，g(x)不变号则<br><span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_a}^{b} f(x)g(x)\, \mathrm{d}x \end{matrix}=f(\xi)\begin{matrix} \int_{_a}^{b}g(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span> <span class="equation-text" contenteditable="false" data-index="1" data-equation="(a\leq\xi\leq b)"><span></span><span></span></span>
积分上限函数<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_a}^{x} f(t)\, \mathrm{d}t \end{matrix}"><span></span><span></span></span>
若<span class="equation-text" data-index="0" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在区间<span class="equation-text" data-index="1" data-equation="[-l,l]" contenteditable="false"><span></span><span></span></span>上连续，则<br>若<span class="equation-text" data-index="2" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>为奇函数，那么<span class="equation-text" contenteditable="false" data-index="3" data-equation="\begin{matrix} \int_{_0}^{x} f(t)\, \mathrm{d}t \end{matrix}"><span></span><span></span></span>为偶函数<br>若<span class="equation-text" data-index="4" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>为偶函数，那么<span class="equation-text" data-index="5" data-equation="\begin{matrix} \int_{_0}^{x} f(t)\, \mathrm{d}t \end{matrix}" contenteditable="false"><span></span><span></span></span>为奇函数<br>
求导
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\begin{matrix} \int_{\phi_1(x)}^{\phi_2(x)} f(t)\, \mathrm{d}t \end{matrix})^\prime=f[\phi_2(x)]\phi^\prime_2(x)-f[\phi_1(x)]\phi^\prime_1(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\begin{matrix} \int_{_{\phi1(x)}}^{\phi_2(x)} g(x-t)f(t)\, \mathrm{d}t \end{matrix})^\prime将g(x-t)中的x和t分离后再进行求导"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\begin{matrix} \int_{_{\phi1(x)}}^{\phi_2(x)} f(x,t)\, \mathrm{d}t \end{matrix})^\prime将f(x,t)中无法分离的部分进行变量代换"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\begin{matrix} \int_{_{\phi1(x)}}^{\phi_2(x)} g(x)f(t)\, \mathrm{d}t \end{matrix})^\prime=g^\prime(x)\begin{matrix} \int_{_{\phi1(x)}}^{\phi_2(x)} f(t)\, \mathrm{d}t \end{matrix}+g(x)(\begin{matrix} \int_{_{\phi1(x)}}^{\phi_2(x)} f(t)\, \mathrm{d}t \end{matrix})^\prime"><span></span><span></span></span>
计算方法
牛顿-莱布尼茨公式<span class="equation-text" data-index="0" data-equation="\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}=F(x)\mid^b_{a}=F(b)-F(a)" contenteditable="false"><span></span><span></span></span>
不定积分里的方法
利用奇偶性和周期性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设f(x)为[-a,a]上的连续函数(a>0)，则"><span></span><span></span></span><br><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{matrix} \int_{_-a}^{a} f(x)\, \mathrm{d}x \end{matrix}=\lbrace^{0,f(x)为奇函数时}_{2\begin{matrix} \int_{_0}^{a} f(x)\, \mathrm{d}x \end{matrix},f(x)为偶函数时}"><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"><span></span><span></span></span><br><br><span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{matrix} \int_{_a}^{a+T} f(x)\, \mathrm{d}x \end{matrix}=\begin{matrix} \int_{_0}^{T} f(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
利用已有公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_0}^{\frac{\pi}{2}} \sin^nx\, \mathrm{d}x \end{matrix}=\begin{matrix} \int_{_0}^{\frac{\pi}{2}} \cos^nx\, \mathrm{d}x \end{matrix}="><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\frac{1}{2}\cdot\frac{\pi}{2}"><span></span><span></span></span> <span class="equation-text" contenteditable="false" data-index="1" data-equation="n为正偶数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\frac{2}{3}"><span></span><span></span></span> <span class="equation-text" contenteditable="false" data-index="1" data-equation="n为大于1的奇数"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_0}^{\pi} xf(\sin x)\, \mathrm{d}x \end{matrix}=\frac{\pi}{2}\begin{matrix} \int_{_0}^{\pi} f(\sin x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
几何方法（求面积）
其他
对于某些积分<span class="equation-text" contenteditable="false" data-index="0" data-equation="I=\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>难以处理时，可以将<span class="equation-text" contenteditable="false" data-index="1" data-equation="I"><span></span><span></span></span>变为另一种形式<span class="equation-text" contenteditable="false" data-index="2" data-equation="I=\begin{matrix} \int_{_a}^{b} g(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span><br>则得<span class="equation-text" contenteditable="false" data-index="3" data-equation="2I=\begin{matrix} \int_{_a}^{b} [f(x)+g(x)]\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>，若该积分易处理，则可使用该方法进行求解
<span class="equation-text" contenteditable="false" data-index="0" data-equation="I=\begin{matrix} \int_{_0}^{b} f(x)\, \mathrm{d}x \end{matrix}，令x=b-t"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="I=\begin{matrix} \int_{_-a}^{a} f(x)\, \mathrm{d}x \end{matrix}，令x=-t"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="I=\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}，令x=b+a-t"><span></span><span></span></span>
分段函数
分段积分法
拼接法
设F(x)是f(x)的一个原函数，即<span class="equation-text" data-index="0" data-equation="\begin{matrix} \int_{}f(x)\, \mathrm{d}x \end{matrix}=F(x)+C" contenteditable="false"><span></span><span></span></span><br>由原函数连续可确定F(x)中的各常数，最后找一点，确定<span class="equation-text" contenteditable="false" data-index="1" data-equation="C"><span></span><span></span></span>的值即可
若f(x)在[a,b]上有界，最多有有限个间断点，则f(x)在[a,b]上可积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_0}^{\pi} \sin^\alpha x\, \mathrm{d}x \end{matrix}=2\begin{matrix} \int_{_0}^{\frac{\pi}{2}} \sin^\alpha x\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
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