积分

[TOC]

定义

定义 定积分作为黎曼和的极限

设f时定义在区间[a,b]的一个函数, 对于[a,b]的任意划分P,设\(c_k\)是在子区间\([x_{k-1},x_k]\)上任意选取的数.

如果存在一个数I,使得不论划分P怎样和\(c_k\)如何选取,都有

\[\lim_{||P|| \to 0}\sum_{k=1}^{n}f(c_k)\Delta x_k = I\]

则称f在[a,b]上是可积的,而I称为f在区间[a,b]上的定积分

\[I=\int_a^bf(x)\mathrm{d}x\]

images

性质

定积分性质
\(\begin{align} &\int_a^bf(x)\mathrm{d}x=-\int_b^af(x)\mathrm{d}x\\ &\int_a^af(x)\mathrm{d}x=0\\ &\int_a^bf(x)\mathrm{d}x=\int_a^cf(x)\mathrm{d}x+\int_c^bf(x)\mathrm{d}x\\ &\int_a^bcf(x)\mathrm{d}x=c\int_a^bf(x)\mathrm{d}x\\ &\int_a^b(f(x)+g(x))\mathrm{d}x=\int_a^bf(x)\mathrm{d}x+\int_a^bg(x)\mathrm{d}x \end{align}\)

微积分基本定理

微积分第一基本定理(不定积分为反导数)

\[\frac{d}{d x}\int_a^xf(t)\mathrm{d}t=f(x)\]

Proof:
\(\begin{align} \frac{d}{d x}\int_a^xf(t)\mathrm{d}t&=\lim_{h \to 0}\frac{\int_a^{x+h}f(t)\mathrm{d}t-\int_a^{x}f(t)\mathrm{d}t}{h}\\ &=\lim_{h \to 0}\frac{\int_h^{x+h}f(t)\mathrm{d}t}{h}\\ &=\lim_{h \to 0}\frac{f(x)h}{h}\\ &=f(x) \end{align}\)

微积分第二基本定理

\[\int_a^bf(x)\mathrm{d}x=F(b)-F(a)\]

Proof:

let \(G(x)=\int_a^xf(t)\mathrm{d}t\), G is one of anti-derivative of f
let \(F(x)=G(x)+C\), F is any of anti-derivative of f
\(\begin{align} F(b)-F(a)&=[G(b)+C]-[G(a)+C]\\ &=G(b)-G(a)\\ &=\int_a^bf(t)dt-\int_a^af(t)dt\\ &=\int_a^bf(t)dt \end{align}\)

求积分方法

积分公式

\[\begin{aligned} &\int d u=u+c\\ &\int k d u=k u+c\\ &\int du + dv=\int d u+\int d v\\ &\int u^{n} d u=\frac{u^{n+1}}{n+1}+C\\ &\int \frac{1}{u} d u=\ln |u|+c\\ &\int \sin u d u=-\cos u+c\\ &\int \cos u d u=\sin u+c\\ &\int \sec u d u=\ln |\sec x+\tan x|+c\\ &\int \csc u d u=-\ln |\csc x+\cot x|+c\\ &\int \sec ^{2} u d u=\tan u+c\\ &\int \csc ^{2} u d u=-\cot u+c\\ &\int \sec u \tan u d u=\sec u+c\\ &\int \csc u \cot u d u=-\csc u+c\\ &\int \tan u d u=\ln |\sec u|+C\\ &\int \cot u d u=\ln |\sin u|+C\\ &\int e^{u} d u=e^{u}+c \\ &\int a^{u} d u=\frac{a^{u}}{\ln a}+c\\ &\int \sinh u d u=\cosh u d x+c\\ &\int \cosh u d u=\sinh u+c\\ &\int \frac{d u}{\sqrt{a^{2}-u^{2}}}=\sin ^{-1}\left(\frac{u}{a}\right)+c\\ &\int \frac{d u}{\sqrt{u^{2}-a^{2}}}=\cos h^{-1}\left(\frac{u}{a}\right)+c \\ &\int \frac{d u}{\sqrt{a^{2}+u^{2}}}=\sinh ^{-1}\left(\frac{u}{a}\right)+c \\ &\int \frac{d u}{a^{2}+u^{2}}=\frac{1}{a} \tan ^{-1}\left(\frac{u}{a}\right)+c\\ &\int \frac{d u}{u \sqrt{u^{2}-a^{2}}}=\frac{1}{a} \sec ^{-1}\left(\frac{u}{a}\right)+C \end{aligned}\]

变量替换

替换积分法

当f和g’是连续函数时,为求积分

\[\int f(g(x))g'(x)dx\]
  1. 做替换\(u=g(x)\),则\(du=g'(x)dx\),得到积分\(\int f(u)du\)
  2. 对u积分
  3. 使用g(x)替代u

定积分的变量替换

\[\int _a^bf(g(x))\cdot g'(x)dx=\int_{g(a)}^{g(b)}f(u)du\]

分部积分

分部积分公式

\[\int u\mathrm{d}v=uv-\int v\mathrm{d}u\]

Proof:
\(\begin{align} \frac{\mathrm{d}}{\mathrm{d}x}(uv)&=u\frac{\mathrm{d}v}{\mathrm{d}x}+v\frac{\mathrm{d}u}{\mathrm{d}x}\\ \int(\frac{\mathrm{d}}{\mathrm{d}x}(uv)\mathrm{d}x&=\int(u\frac{\mathrm{d}v}{\mathrm{d}x})+\int (v\frac{\mathrm{d}u}{\mathrm{d}x})\\\int u\mathrm{d}v&=uv-\int v\mathrm{d}u \end{align}\)

列表积分

对于形如\(\int f(x)g(x)dx\)的积分, 其中f(x)可以反复求导得到0, g(x)可以简单地重复积分


1.15

列表积分2

1.7

部分分式

形如\(\int \frac{dx}{x^2+px+q}\)

\(\Delta>0\):

\[\frac{1}{x^2+px+q}=\frac{A_1}{x-a_1}+\frac{A_2}{x-a_2}\\\]

其中

\[A_1(x-a_2)+A_2(x-a_1)=1\]

\(\Delta=0\):

\[\int \frac{dx}{x^2+px+q}=\int \frac{dx}{(x-a)^2}=-\frac{1}{x-a}+C\]

\(\Delta<0\):

配方

\[x^2+px+q=(x+\frac{p}{2})^2-\frac{\Delta}{4}\]

化简

\[\begin{align} \int \frac{dx}{x^2+px+q}=&-\frac{4}{\Delta}\int \frac{dx}{1+(\frac{2x+p}{\sqrt{-\Delta}})^2}\\ =&\frac{2}{\sqrt{-\Delta}}\int\frac{du}{u^2+1}\\ =&\frac{2}{\sqrt{-\Delta}}tan^{-1}(\frac{2x+p}{\sqrt{-\Delta}})+C \end{align}\]

形如\(\int\frac{mx+n}{x^2+px+q}dx\)

\(\Delta>0\):

\[\frac{mx+n}{x^2+px+q}=\frac{A_1}{x-a_1}+\frac{A_2}{x-a_2}\]

\(\Delta=0\):

\[\frac{mx+n}{x^2+px+q}=\frac{A}{x-a}+\frac{B}{(x-a)^2}\]

\(\Delta<0\):

换元后长除

\[u=x^2+px+q\\ du=2x+p\\ \int\frac{mx+n}{x^2+px+q}dx=\int (r\frac{2x+p}{x^2+px+q}+s\frac{1}{x^2+px+q})dx\\ mx+n=r(2x+q)+s\\\]

三角换元

三角替换 \(x=atan\theta\\ a^2+x^2=a^2sec^2\theta\)

\[x=asin\theta\\ a^2-x^2=a^2cos^2\theta\] \[x=asec\theta\\ x^2-a^2=a^2tan^2\theta\]

估算积分

1.黎曼和

在区间[a,b],均分为n等分\(h=\frac{b-a}{n}\)

\[\int_a^bf(x)dx\approx h\sum_{j=1}^{n}f(c_j)\]

2.梯形法则

在区间[a,b],均分为n等分\(h=\frac{b-a}{n}\)

\[\int_a^bf(x)dx\approx \frac{h}{2}\sum_{j=1}^{n}(f(x_{j-1})+f(x_j))\]

3.辛普森法则(二次曲线逼近)

在区间[a,b],均分为n等分\(h=\frac{b-a}{n}\)

\[\int_a^bf(x)dx\approx \frac{h}{3}\sum_{j=1}^{\frac{1}{2}n}(f(x_{2j-2})+4f(x_{2j-1})+f(x_{2j}))\]

Proof:

\(A=\frac{h}{3}(y_0+4y_1+y_2)\) 1.9