CAL-4-不定积分
2021-08-01 14:48:17 0 举报AI智能生成
高等数学微积分第四章 不定积分 知识点梳理
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大纲/内容
基本初等函数的不定积分
<span class="equation-text" data-index="0" data-equation="\int k \mathrm dx=kx+C" contenteditable="false"><span></span><span></span></span>
幂，指<br>
<span class="equation-text" data-index="0" data-equation="\int x\mathrm dx=\frac 1 2x^2+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int x^a\mathrm dx=\frac {x^{a+1}} {a+1}+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int e^x \mathrm dx=e^x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int a^x \mathrm dx=\frac{a^x}{\ln{a}}+C" contenteditable="false"><span></span><span></span></span>
三角函数<br>
<span class="equation-text" data-index="0" data-equation="\int\sin x\mathrm dx=-\cos x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\cos x\mathrm dx=\sin x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\tan x\mathrm dx=-\ln{|\cos x|}+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\cot x\mathrm dx=\ln{|\sin x|}+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\sec x\mathrm dx=\ln|\sec x+\tan x|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\csc x\mathrm dx=\ln|\csc x-\cot x|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\sec^2x\mathrm dx=\tan x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\csc^2x\mathrm dx=-\cot x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\sec x\tan x\mathrm dx=\sec x+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\csc x\cot x\mathrm dx=-\csc x+C" contenteditable="false"><span></span><span></span></span>
平方和/差、<br>根号平方和/差<br>三角代换<br>
<span class="equation-text" data-index="0" data-equation="\int\frac{\mathrm dx}{x^2+a^2}=\frac 1 a\arctan{\frac x a}+C(a\neq0)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\frac{\mathrm dx}{x^2-a^2}=\frac 1 {2a}\ln|\frac {x-a}{x+ a}|+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\frac{\mathrm dx}{\sqrt{a^2-x^2}}=\arcsin{\frac x a}+C(a>0)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int{\sqrt{a^2-x^2}}\mathrm dx=\frac x 2\sqrt{a^2-x^2}+\frac{a^2}2\arcsin\frac x a+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\frac{\mathrm dx}{\sqrt{x^2+a^2}}=\ln(x+\sqrt{x^2+a^2})+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\frac{\mathrm dx}{\sqrt{x^2-a^2}}=\ln(x+\sqrt{x^2-a^2})+C" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\sqrt{a^2+x^2}\mathrm dx=\frac x 2\sqrt{a^2+x^2}+\frac{a^2}2\ln|x+\sqrt{a^2+x^2}|+C" contenteditable="false"><span></span><span></span></span>
不定积分方法
第一类换元积分法（凑微分）
核心原理
确定φ(x)是关键
第二类换元积分法
核心原理
通常f[φ(t)]φ'(t)积分比较简单
应用
平方和、平方差
√(a²-x²)
令x=asint
=acost
√(x²+a²)
令x=atant
=asect
√(x²-a²)
令x=asect
=atant
分部积分法
核心原理
观察发现，等式右边求导的只有u
任何积分都可以分部积分，可以直接相当于<br><span class="equation-text" data-index="0" data-equation="\int u\mathrm{d}x = xu-\int x\mathrm{d}u" contenteditable="false"><span></span><span></span></span>
用法
幂指相乘
正幂次：用指数构造dv
结论：<span class="equation-text" data-index="0" data-equation="\int x^ie^{kx}\mathrm dx=\frac{e^{kx}}{k}(x^i-\frac{ix^{i-1}}{k}+\frac{i(i-1)x^{i-2}}{k^2}-\frac{i(i-1)(i-2)x^{i-3}}{k^3}+...)+C" contenteditable="false"><span></span><span></span></span>
负幂次：用指数构造dv<br>
结论：<span class="equation-text" data-index="0" data-equation="\int x^ie^{kx}\mathrm dx=e^{kx}(\frac{x^{i+1}}{i+1}-\frac{kx^{i+2}}{(i+1)(i+2)}+\frac{k^2x^{i+3}}{(i+1)(i+2)(i+3)}...)+C" contenteditable="false"><span></span><span></span></span><br>
例：求<span class="equation-text" data-index="0" data-equation="xy''+2y'=e^x" contenteditable="false"><span></span><span></span></span>通解
幂对相乘
用幂函数构造dv
幂函数*三角
用三角函数构造dv
幂函数*反三角
用幂函数构造dv
指数*三角
均可构造
注意：出现原不定积分，要移项
<span class="equation-text" data-index="0" data-equation="\int\sec^nx\mathrm dx" contenteditable="false"><span></span><span></span></span> 或 <span class="equation-text" data-index="1" data-equation="\int\csc^nx\mathrm dx" contenteditable="false"><span></span><span></span></span>
n为偶数：提一个sec²x
<span class="equation-text" data-index="0" data-equation="\rightarrow\int\sec^{n-2}x\mathrm d\tan x=\int(\tan^2 x+1)^{\frac n 2-1}\mathrm d\tan x=\int(x^2+1)^{\frac n 2-1}\mathrm d x" contenteditable="false"><span></span><span></span></span>
n为奇数：提一个sec²x
<span class="equation-text" data-index="0" data-equation="\rightarrow\int\sec^{n-2}x\mathrm d\tan x=\sec^{n-2}x\tan x-\int\tan x\mathrm d\sec^{n-2}x" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="=sec^{n-2}x\tan x-(n-2)\int\sec^{n-3}x\sec x\tan x\tan x\mathrm dx" contenteditable="false"><span></span><span></span></span><br>
<span class="equation-text" data-index="0" data-equation="=sec^{n-2}x\tan x-(n-2)\int\sec^{n-2}x(\sec^2x-1)\mathrm dx" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\sec^nx\mathrm dx=sec^{n-2}x\tan x-(n-2)\int\sec^nx-\sec^{n-2}x\mathrm dx" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\int\sec^nx\mathrm dx=\frac{sec^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int\sec^{n-2}x\mathrm dx" contenteditable="false"><span></span><span></span></span>
注意：出现原不定积分，要移项
有理函数不定积分
将R(x)化为真分式
将分母因式分解，再化成部分和
(Ax+B)
贡献一个1/(Ax+B)
(Ax+B)^n
从1到n，贡献n个i/(Ax+B)^i
(Ax²+Bx+C)
贡献一个(Ax+B)/((Ax²+Bx+C))
三角函数微分学<br>
<span class="equation-text" data-index="0" data-equation="\sin' x=\cos x" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\cos' x=-\sin x" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\tan' x=\sec^2 x" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\cot' x=-\csc^2 x" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\sec' x=\sec x\tan x" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\csc' x=-\csc x\cot x" contenteditable="false"><span></span><span></span></span>
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