积分是整个高等数学的核心内容
积分概念 & 性质
性质
线性性质：$\int(k_1f_1(x)+k_2f_2(x))=k_1\int f_1(x)+k_2 \int f_2(x)$
定积分存在定理：连续、单调、有界且有限个间断点
定积分必要条件：定积分必有界
可加性：$\int _a^b=\int _a^c+\int _c^b$
估值定理：$m(b-a)≤\int_a^b f(x)≤M(b-a)$
几何意义
$\int_a^bf(x)\text{d}x=F(b)-F(a)$
$\int_{x_0}^xf’(t)\text{d}t=f(x)-f(x_0)$
$\int_{-a}^a{f\left( x \right) \text{d}x=\left\{ \begin{array}{l} 2\int_0^a{f\left( x \right) \text{d}x},\ f\left( x \right) =f\left( -x \right)\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ,\ f\left( x \right) =-f\left( -x \right)\\ \end{array} \right.}$保号性
- 若$f(x)≤g(x)$，则$\int f(x)\text{d}x≤\int g(x)\text{d}x$
- $|\int f(x)\text{d}x|≤\int|f(x)|\text{d}x$
- 若$f(x)$非负且不恒等于0，则$\int f(x) dx>0$
反常积分收敛
无穷反常积分：$\int_a^∞ f(x)\text{d}x$
瑕积分：$\int_a^bf(x)\text{d}x$，其中$\underset{x\rightarrow a^+}{\lim}f(x)=∞$
计算简单积分
比较审敛法
对于$\int_a^{+\infty}f(x)\text dx$，则$\underset{x\rightarrow \infty}{\lim}\dfrac{f\left( x \right)}{g\left( x \right)}\text{d}x=l\left\{ \begin{array}{l} \text{大收小收 }\ \ \ \ \ \ \ \ \ \ \ l=0\\ \text{小散大散 }\ \ \ \ \ \ \ \ \ \ \ l=+\infty\\ \text{同敛散性 } \ \ \ \ 0 < l<+\infty\\ \end{array} \right. $
瑕积分可考虑构造$g(x)=\dfrac1{P_m(x-a)}$，则$\underset{x\rightarrow a^+}{\lim}\dfrac{f(x)=\dfrac1{P_m(x-a)h(x)}}{g(x)}=\dfrac1{h(a)}$
极限审敛法
$\infty$反常积分：$\underset{x\rightarrow \infty}{\lim}x^{\rho}f\left( x \right) =\text{C}\left\{ \begin{array}{l} \rho >1\text{收敛}\\ \rho \le 1\text{发散}\\ \end{array} \right. $
瑕积分：$\underset{x\rightarrow a^+}{\lim}x^{\rho}f\left( x \right) =\text{C}\left\{ \begin{array}{l} \rho <1\text{收敛}\\ \rho \ge 1\text{发散}\\ \end{array} \right. $
对于$\int\dfrac{\text dx}{x^\alpha\ln x^\beta}$
瑕积分：$\left\{\begin{array}{}\alpha <1\\\alpha=1,\ \beta>1\end{array}\right.$收敛
反常积分：$\left\{\begin{array}\\ \alpha>1 \\ \alpha = 1,\ \beta>1 \end{array}\right.$收敛
计算
不定积分
基本积分公式
指对幂函数
$\int{x^k}\text{d}x=\dfrac{1}{k+1}x^{k+1}+\text{C}$
$\int{\dfrac{1}{x}}\text{d}x=\ln\text{|}x|+\text{C}$
$\int{\dfrac{1}{x^2}}\text{d}x=-\dfrac{1}{x}+\text{C}$
$\int{\dfrac{1}{\sqrt{x}}}\text{d}x=2\sqrt{x}+\text{C}$
$\int{\text{e}^x}\text{d}x=\text{e}^x+\text{C}$
$\int{a^x}\text{d}x=\dfrac{a^x}{\ln a}+\text{C}$
$\ln x\text dx=x\left(\ln x-1\right)+\text C$
$\int (ax+b)^m\text dx=\dfrac1{a(m+1)}(ax+b)^{m+1}+\text C$
$\int\dfrac{\text dx}{(x-a)^k}\text dx=\dfrac{(x-a)^{1-k}}{1-k}+\text C$
$\int{\dfrac{1}{1+x^2}}\text{d}x=\arctan x+\text{C}$
$\int{\dfrac{1}{a^2+x^2}}\text{d}x=\dfrac{1}{a}\arctan \dfrac{x}{a}+\text{C}$
$\int{\dfrac{1}{\sqrt{a^2+x^2}}}\text{d}x=\ln \left( x+\sqrt{x^2+a^2} \right) +\text{C}$
$\int{\dfrac{1}{\sqrt{1-x^2}}}\text{d}x=\arcsin x+\text{C}$
$\int{\dfrac{1}{\sqrt{a^2-x^2}}}\text{d}x=\arcsin \dfrac{x}{a}+\text{C}$
$\int{\dfrac{1}{\sqrt{x^2-a^2}}}\text{d}x=\ln\text{|}x+\sqrt{x^2+a^2}|+\text{C}$
$\int{\dfrac{1}{x^2-a^2}}\text{d}x=\dfrac{1}{2a}\ln\text{|}\dfrac{x-a}{x+a}|+\text{C}$
$\int{\dfrac{1}{a^2-x^2}}\text{d}x=\dfrac{1}{2a}\ln\text{|}\dfrac{x+a}{x-a}|+\text{C}$
$\int{\sqrt{a^2-x^2}}\text{d}x=\dfrac{a^2}{2}\arcsin \dfrac{a}{x}+\dfrac{x}{2}\sqrt{a^2-x^2}+\text{C}$
$\int{\dfrac{Mx+N}{x^2+px+q}}\text{d}x=\dfrac{M}{2}\ln \left( x^2+px+q \right) +\dfrac{2N-Mp}{\sqrt{4q-p^2}}\arctan \dfrac{2x+p}{\sqrt{4q-p^2}}+\text{C}$
三角函数
$\int{\sin}x\text{d}x=-\cos x+\text{C}$
$\int{\cos}x\text{d}x=\sin x+\text{C}$
$\int{\tan}x\text{d}x=-\ln\text{|}\cos x|+\text{C}=\ln\text{|}\sec x|+\text{C}$
$\int{\csc}x\text{d}x=\ln\text{|}\csc x-\cot x|+\text{C}$
$\int{\sec}x\text{d}x=\ln\text{|}\sec x+\tan x|+\text{C}$
$\int{\cot}x\text{d}x=\ln\text{|}\sin x|+\text{C}$
$\int{\arcsin x\text{d}x}=x\arcsin x+\sqrt{1-x^2}+\text{C}$
$\int{\arccos x\text{d}x}=x\arccos x-\sqrt{1-x^2}+\text{C}$
常见积分
$\int x\text e^{-x}\text dx=-(x+1)\text e^{-x}$
$\int\ln(1+x)\text dx=x\ln(1+x)-x+\ln(1+x)$
$\int{\sec}x\tan x\text{d}x=\sec x+\text{C}$
$\int{\csc}x\cot x\text{d}x=-\csc x+\text{C}$
$\int{\sin ^2x}\text{d}x=-\dfrac{\sin \left( 2x \right) -\left( 2x \right)}{4}$
$\int{\cos ^2x\text{d}x}=\dfrac{\sin \left( 2x \right) +\left( 2x \right)}{4}$
$\int{\tan ^2x\text{d}x}=\tan x-x$
$\int{\csc ^2x}\text{d}x=-\dfrac{1}{\tan x}$
$\int{\sec ^2x}\text{d}x=\tan x$
$\int{\cot ^2x}\text{d}x=-x-\dfrac{1}{\tan x}$
$\int\dfrac1{\sin^{n+2}x}\text dx=\dfrac n{n+1}\int\dfrac1{\sin^nx}\text dx-\dfrac1{n+1}\dfrac{\cos x}{\sin^{n+1}x}$
$\dfrac1{\sin^2ax\cdot\cos^2ax}\text dx=-\dfrac 2a\cot(2ax)+\text C$
$\int \text e^{ax}\cos bx\ \text dx=\dfrac{\text e^{ax}(a\cos bx+b\sin bx)}{a^2+b^2}+\text C$
$\int \text e^{ax}\sin bx\ \text dx=\dfrac{\text e^{ax}(a\sin bx-b\cos bx)}{a^2+b^2}+\text C$
积分方式
换元
三角函数代换
$\sqrt{a^2-x^2}\rightarrow x=a\sin t，|t|<\dfrac{\pi}{2}$，$\text dx=a\cos t\text dt$，$\sqrt{a^2-x^2}=\cos t$
$\sqrt{a^2+x^2}\rightarrow x=a\tan t，|t|<\dfrac{\pi}{2}$，$\text dx=a\sec^2 t\text dt$，$\sqrt{a+x^2}=a \sec t$
$\sqrt{x^2-a^2}\rightarrow x=a\sec t$，若$x>0$，则$\left\{ \begin{array}{l} x>0\text{，则}0
恒等变形后三角代换
当含有$\sqrt{ax^2+bx+c}$时，可化为$\sqrt{\varphi^2(x)+k^2}$、$\sqrt{\varphi^2(x)-k^2}$、$\sqrt{k^2-\varphi^2(x)}$，然后三角代换根式代换
$\sqrt{\dfrac{ax+b}{cx+d}}\sqrt{ae^x+c}\sqrt[n]{ax+b}$不方便凑平方项的直接令$t=\sqrt{x}$
倒代换
分母幂次比分子高两次以上，令$t=\dfrac{1}{x}$
复杂函数代换
$ (1-\dfrac1{x^2})f(x+\dfrac1x)\text dx= f(x+\dfrac1x)\text d(x+\dfrac1x)$
$(1+\dfrac1{x^2})f(x-\dfrac1x)\text dx= f(x-\dfrac1x)\text d(x-\dfrac1x)$
$(\text e^x+\text e^{-x})\text dx=\text d(\text e^x-\text e^{-x})$
$(\text e^x-\text e^{-x})\text dx=\text d(\text e^x+\text e^{-x})$
$\int\dfrac{f(x^n)}x \text dx=\dfrac1n\int\dfrac {f(x^n)}{x^n}\text{d}(x^n)$
分部积分
$\int udv=uv-\int vdu$
$\underset{u\longleftrightarrow v}{反对幂指三}$
$\sec^nx$与$\csc^n x$
分式积分
- 代入法计算参数
将分子或分母尝试求导
对于分式拆项
实在没办法：$\dfrac 1{x-a}\ \ \ \ \dfrac{Ax+B}{x^2+px+q}$
分子对着分母拆项
对于$\sin x,\ \cos x$分式，巧用$\sin^2x+\cos^2x=1$
对于$\int \dfrac{\text dx}{a\sin x+b\cos x}$，使其为$\dfrac 1{\sqrt{a^2+b^2}}\int\dfrac{\text dx}{\sin(\varphi x+\alpha)}$
对于$\sqrt{ax^2+bx+c}$化为$\sqrt{a^2±x^2}$或$\sqrt{x^2+\alpha^2}$
分母分子同乘一个因子
分母为$1+\sin x/1+\cos x$，同乘$1-\sin x/1-\cos x $
对于$\text e^x$凑$\int f(\text e^x)\text d e^x$
对于隐函数，尝试参数方程
形如$\int\dfrac{M\sin x+N\cos x}{A\sin x+B\cos x}\text dx$
可用待定系数凑为$\dfrac{X(A\sin x+B\cos x)+Y(A\sin x+B\cos x)’}{A\sin x+B\cos x}$
三角函数积分
- $1=\sin^2x+\cos^2x=\sec^2x-\tan^2x$
对于$\dfrac 1{1±\bigtriangleup }$，考虑共轭分式
对于$\bigtriangleup +\text C$，使用万能公式
令$t=\tan\dfrac x2$
$\sin x=\dfrac{2t}{1+t^2}$
$\cos x=\dfrac{1-t^2}{1+t^2}$
$\text dx=\dfrac{2\text dt}{1+t^2}$
$\sin x+\cos x$，考虑辅助角
$\int \csc x\text dx=\ln |\csc x-\cot x|\text dx$
$\int \sec x\text dx=\ln|\sec x+\tan x|\text dx$
$\int \cot^2 x\text dx=-\cot x-x+\text C$
$\int \tan^4x\text dx=\dfrac13 \tan^3x-\tan x+\text C$
- $\text d(\sin^2x)=2\sin x\cos x=\sin 2x\text dx$
cos x\sin x奇次 偶次 0次 奇次 将 cos x放d后，剩余的转换为sin xsinmx·cosnx dx
=sinmx·cosn-1x dsinx偶次 sinmx·cosnx dx
=-sinm-1x·cosnx dcosxsin x→cos x,转⑥cos x→sin x，转⑧倍角公式 0次 倍角公式 / $\int \tan^n x\text dx=\dfrac{\tan^{n-1}x}{n-1}-\int\tan ^{n-2}x\text dx$
$\int\dfrac{\text dx}{a\sin x+b\cos x}\text dx=\dfrac1{\sqrt{a^2+b^2}}\ln\left|\tan x-\dfrac{x+\arctan \dfrac ba}{2}\right|$
定积分计算
$f(x)$为偶函数$\Rightarrow\int_{-a}^af(x)dx=2\int_0^af(x)dx$
$f(x)$为奇函数$\Rightarrow\int_{-a}^af(x)dx=0$
$f(x)$为周期函数$\Rightarrow\int_a^{a+T}f(x)dx=\int_0^Tf(x)dx$
区间再现公式：
$\int_a^b f(x)dx=\int_a^bf(a+b-x)dx$原函数分部积分不适合$\int_a^b f(x)\text dx=(b-a)\int_0^1 f[a+(b-a)x]\text dx$
华理士公式：$\int _0^{\dfrac{\pi}{2}}\sin^nxdx=\int _0^{\dfrac{\pi}{2}}\cos^nxdx= \left\{ \begin{array}{l} \dfrac{n-1}{n}\cdot\dfrac{n-3}{n-2}\cdot\cdot\cdot\cdot\dfrac{2}{3}\cdot1\text{，}n\text{为奇数}\\ \dfrac{n-1}{n}\cdot\dfrac{n-3}{n-2}\cdot\cdot\cdot\cdot\dfrac{1}{2}\cdot \dfrac{\pi}{2}\text{，}n\text{为偶数}\\ \end{array} \right.$
$\int_0^1{x^m\left( 1-x \right) ^n}\text{d}x=\int_0^1{x^n\left( 1-x \right) ^m}\text{d}x$
$\int_{-a}^a{f\left( x \right)}\text{d}x=\int_0^a{\left[ f\left( x \right) +f\left( -x \right) \right]}\text{d}x$
$\int_0^{\frac{\pi}{2}}{f\left( \sin x,\cos x \right)}\text{d}x=\int_0^{\frac{\pi}{2}}{f\left( \cos x,\sin x \right)}\text{d}x$
$\int_0^{\pi}{xf\left( \sin x \right)}\text{d}x=\dfrac{\pi}{2}\int_0^{\pi}{f\left( \sin x \right)}\text{d}x=\pi\int^{\frac\pi2}_0f(\sin x)\text dx$
变限积分求导
若$F(x)=\int_{\varphi_1(x)}^{\varphi_2(x)}f(t)dt$，则$F’(x)=f[\varphi_2(x)]\varphi’_2(x)-f[\varphi_1(x)]\varphi’_1(x)$
性质：
- $f(x)\in T\Rightarrow\underset{x\rightarrow \infty}{\lim}\dfrac{\int_0^x{f\left( t \right)}\text{d}t}{x}=\dfrac{\int_0^T{f\left( t \right)}\text{d}t}{T}$
- 广义积分仅可在收敛时才可使用奇偶性
定积分换元
| 原式 | $F(x)$ | $F’(X)$ | $F’’(x)$ |
|---|---|---|---|
| $\int_0^xf(x-t)\text dt$ | $\int_0^xf(u)\text du$ | $f(x) $ | $f’(x)$ |
| $\int_0^x(x-t)f(t)\text dt\\int_0^x tf(x-t)\text d t$ | $x\int_0^xf(t)\text dt-\int_0^xtf(t)\text dt$ | $\int_0^xf(t)\text dt$ | $f(x )$ |
| $\int_0^1xf(tx)\text dt$ | $\int_0^x f(u)\text du$ | $f(x) $ | $f’(x) $ |
| $\int_0^xtf(t^2-x^2)\text dt$ | $-\dfrac12\int_0^{x^2}f(u)\text du$ | $-xf(x^2)$ |
