附录 5-6 复数与简明积分表 · 《工科数学分析基础（第三版）》

附录 5 复数及其运算

一、复数的概念

我们在实数范围内，方程

x^2 = -1 \tag{5.1}

是无解的，因为没有一个实数的平方等于 $-1$ 。由于解方程的需要，人们引进一个新数 $i$ ，称为虚数单位，并规定

i^2 = -1,

从而 $i$ 是方程 $x^2 = -1$ 的一个根。

复数的定义

对于任意两个实数 $x, y$ ，我们称 $z = x + yi$ 为复数，其中 $x, y$ 分别称为 $z$ 的实部和虚部，记作

x = \text{Re}(z), \quad y = \text{Im}(z).

当 $x = 0, y \neq 0$ 时， $z = iy$ 称为纯虚数；当 $y = 0$ 时， $z = x + 0i$ ，我们把它看作是实数 $x$ 。

两个复数相等，必须且只需它们的实部和虚部分别相等。一个复数 $z$ 等于 $0$ ，必须且只需它的实部和虚部同时等于 $0$ 。

二、复数的代数运算

两个复数 $z_1 = x_1 + iy_1, z_2 = x_2 + iy_2$ 的加法、减法及乘法定义如下：

(x_1 + iy_1) \pm (x_2 + iy_2) = (x_1 \pm x_2) + i(y_1 \pm y_2),(x_1 + iy_1)(x_2 + iy_2) = (x_1x_2 - y_1y_2) + i(x_2y_1 + x_1y_2),

并分别称以上两式右端的复数为 $z_1$ 与 $z_2$ 的和、差与积。

我们又称满足

z_2 z = z_1 \quad (z_2 \neq 0)

的复数 $z = x + iy$ 为 $z_1$ 除以 $z_2$ 的商，记作 $z = \frac{z_1}{z_2}$ 。从这个定义立即可推得

z = \frac{z_1}{z_2} = \frac{x_1x_2 + y_1y_2}{x_2^2 + y_2^2} + i \frac{x_2y_1 - x_1y_2}{x_2^2 + y_2^2}.

不难证明，与实数的情形一样，复数的运算也满足交换律、结合律和分配律：

z_1 + z_2 = z_2 + z_1, \quad z_1 z_2 = z_2 z_1,z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3, \quad z_1(z_2 z_3) = (z_1 z_2)z_3,z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3.

共轭复数

我们把实部相同而虚部绝对值相等而符号相反的两个复数称为共轭复数，与 $z$ 共轭的复数记为 $\bar{z}$ 。如果 $z = x + iy$ ，那么 $\bar{z} = x - iy$ 。共轭复数有如下性质：

$\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2, \quad \overline{z_1 z_2} = \bar{z}_1 \bar{z}_2, \quad \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}$
$z\bar{z} = [\text{Re}(z)]^2 + [\text{Im}(z)]^2$
$z + \bar{z} = 2\text{Re}(z), \quad z - \bar{z} = 2i\text{Im}(z)$

三、复数的三种表示式及 Euler 公式

代数表示式：

$z = x + iy.$

三角表示式：

$z = r(\cos\theta + i\sin\theta).$

指数表示式：

$z = re^{i\theta}.$

Euler 公式：

$e^{i\theta} = \cos\theta + i\sin\theta,$

由此可得

$\cos\theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta}), \quad \sin\theta = \frac{1}{2i}(e^{i\theta} - e^{-i\theta}).$

附录 6 简明积分表

本附录汇集了工科数学分析中最常用的不定积分公式，其中未标明积分常数 $C$ 的公式均应加上任意常数 $C$ 。

一、含有 $a+bx$ 的积分

\int \frac{\mathrm{dx}}{x(a+bx)^2} = \frac{1}{a(a+bx)} - \frac{1}{a^2}\ln\left|\frac{a+bx}{x}\right| + C \tag{8}

二、含有 $\sqrt{a+bx}$ 的积分

\int x\sqrt{a+bx} \, \mathrm{dx} = \frac{2(3bx-2a)(a+bx)^{3/2}}{15b^2} + C \tag{9}

\int x^2\sqrt{a+bx} \, \mathrm{dx} = \frac{2(15b^2x^2 - 12abx + 8a^2)(a+bx)^{3/2}}{105b^3} + C \tag{10}

\int \frac{x \, \mathrm{dx}}{\sqrt{a+bx}} = \frac{2(bx-2a)\sqrt{a+bx}}{3b^2} + C \tag{11}

\int \frac{x^2 \, \mathrm{dx}}{\sqrt{a+bx}} = \frac{2(3b^2x^2 - 4abx + 8a^2)\sqrt{a+bx}}{15b^3} + C \tag{12}

\int \frac{\mathrm{dx}}{x\sqrt{a+bx}} = \begin{cases} \displaystyle \frac{1}{\sqrt{a}}\ln\left|\frac{\sqrt{a+bx}-\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right| + C, & \text{当 } a > 0, \\ \displaystyle \frac{2}{\sqrt{-a}}\arctan\sqrt{\frac{a+bx}{-a}} + C, & \text{当 } a < 0. \end{cases} \tag{13}

\int \frac{\mathrm{dx}}{x^2\sqrt{a+bx}} = -\frac{\sqrt{a+bx}}{ax} - \frac{b}{2a}\int \frac{\mathrm{dx}}{x\sqrt{a+bx}} \tag{14}

\int \frac{\sqrt{a+bx}}{x} \, \mathrm{dx} = 2\sqrt{a+bx} + a\int \frac{\mathrm{dx}}{x\sqrt{a+bx}} \tag{15}

\int \frac{\sqrt{a+bx}}{x^2} \, \mathrm{dx} = -\frac{\sqrt{a+bx}}{x} + \frac{b}{2}\int \frac{\mathrm{dx}}{x\sqrt{a+bx}} \tag{16}

三、含有 $a^2 \pm x^2$ 的积分

\int \frac{\mathrm{dx}}{(a^2+x^2)^n} = \begin{cases} \displaystyle \frac{1}{a}\arctan\frac{x}{a} + C, & \text{当 } n = 1, \\ \displaystyle \frac{x}{2(n-1)a^2(a^2+x^2)^{n-1}} + \frac{2n-3}{2(n-1)a^2}\int \frac{\mathrm{dx}}{(a^2+x^2)^{n-1}}, & \text{当 } n > 1. \end{cases} \tag{17}

\int \frac{x \, \mathrm{dx}}{(a^2+x^2)^n} = \begin{cases} \displaystyle \frac{1}{2}\ln(a^2+x^2) + C, & \text{当 } n = 1, \\ \displaystyle -\frac{1}{2(n-1)(a^2+x^2)^{n-1}} + C, & \text{当 } n > 1. \end{cases} \tag{18}

\int \frac{\mathrm{dx}}{a^2-x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C \tag{19}

四、含有 $\sqrt{a^2-x^2}$ 的积分

\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 \tag{20}

\int x\sqrt{a^2-x^2} \, \mathrm{dx} = -\frac{1}{3}(a^2-x^2)^{3/2} + C \tag{21}

\int x^2\sqrt{a^2-x^2} \, \mathrm{dx} = \frac{x}{8}(2x^2-a^2)\sqrt{a^2-x^2} + \frac{a^4}{8}\arcsin\frac{x}{a} + C \tag{22}

\int \frac{\mathrm{dx}}{\sqrt{a^2-x^2}} = \arcsin\frac{x}{a} + C \tag{23}

\int \frac{x \, \mathrm{dx}}{\sqrt{a^2-x^2}} = -\sqrt{a^2-x^2} + C \tag{24}

\int \frac{x^2 \, \mathrm{dx}}{\sqrt{a^2-x^2}} = -\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin\frac{x}{a} + C \tag{25}

\int (a^2-x^2)^{3/2} \, \mathrm{dx} = \frac{x}{8}(5a^2-2x^2)\sqrt{a^2-x^2} + \frac{3a^4}{8}\arcsin\frac{x}{a} + C \tag{26}

\int \frac{\mathrm{dx}}{(a^2-x^2)^{3/2}} = \frac{x}{a^2\sqrt{a^2-x^2}} + C \tag{27}

\int \frac{x \, \mathrm{dx}}{(a^2-x^2)^{3/2}} = \frac{1}{\sqrt{a^2-x^2}} + C \tag{28}

\int \frac{x^2 \, \mathrm{dx}}{(a^2-x^2)^{3/2}} = \frac{x}{\sqrt{a^2-x^2}} - \arcsin\frac{x}{a} + C \tag{29}

\int \frac{\mathrm{dx}}{x\sqrt{a^2-x^2}} = \frac{1}{a}\ln\left|\frac{a-\sqrt{a^2-x^2}}{x}\right| + C \tag{30}

\int \frac{\mathrm{dx}}{x^2\sqrt{a^2-x^2}} = -\frac{\sqrt{a^2-x^2}}{a^2x} + C \tag{31}

\int \frac{\mathrm{dx}}{x^3\sqrt{a^2-x^2}} = -\frac{\sqrt{a^2-x^2}}{2a^2x^2} - \frac{1}{2a^3}\ln\left|\frac{a+\sqrt{a^2-x^2}}{x}\right| + C \tag{32}

\int \frac{\sqrt{a^2-x^2}}{x} \, \mathrm{dx} = \sqrt{a^2-x^2} - a\ln\left|\frac{a+\sqrt{a^2-x^2}}{x}\right| + C \tag{33}

\int \frac{\sqrt{a^2-x^2}}{x^2} \, \mathrm{dx} = -\frac{\sqrt{a^2-x^2}}{x} - \arcsin\frac{x}{a} + C \tag{34}

五、含有 $\sqrt{x^2 \pm a^2}$ 的积分

\int \sqrt{x^2\pm a^2} \, \mathrm{dx} = \frac{x}{2}\sqrt{x^2\pm a^2} \pm \frac{a^2}{2}\ln\left|x+\sqrt{x^2\pm a^2}\right| + C \tag{35}

\int x\sqrt{x^2\pm a^2} \, \mathrm{dx} = \frac{1}{3}(x^2\pm a^2)^{3/2} + C \tag{36}

\int x^2\sqrt{x^2\pm a^2} \, \mathrm{dx} = \frac{x}{8}(2x^2\pm a^2)\sqrt{x^2\pm a^2} - \frac{a^4}{8}\ln\left|x+\sqrt{x^2\pm a^2}\right| + C \tag{37}

\int \frac{\mathrm{dx}}{\sqrt{x^2\pm a^2}} = \ln\left|x+\sqrt{x^2\pm a^2}\right| + C \tag{38}

\int \frac{x \, \mathrm{dx}}{\sqrt{x^2\pm a^2}} = \sqrt{x^2\pm a^2} + C \tag{39}

\int \frac{x^2 \, \mathrm{dx}}{\sqrt{x^2\pm a^2}} = \frac{x}{2}\sqrt{x^2\pm a^2} \mp \frac{a^2}{2}\ln\left|x+\sqrt{x^2\pm a^2}\right| + C \tag{40}

\int (x^2\pm a^2)^{3/2} \, \mathrm{dx} = \frac{x}{8}(2x^2\pm 5a^2)\sqrt{x^2\pm a^2} \pm \frac{3a^4}{8}\ln\left|x+\sqrt{x^2\pm a^2}\right| + C \tag{41}

\int \frac{\mathrm{dx}}{(x^2\pm a^2)^{3/2}} = \pm \frac{x}{a^2\sqrt{x^2\pm a^2}} + C \tag{42}

\int \frac{x \, \mathrm{dx}}{(x^2\pm a^2)^{3/2}} = -\frac{1}{\sqrt{x^2\pm a^2}} + C \tag{43}

\int \frac{x^2 \, \mathrm{dx}}{(x^2\pm a^2)^{3/2}} = -\frac{x}{\sqrt{x^2\pm a^2}} + \ln\left|x+\sqrt{x^2\pm a^2}\right| + C \tag{44}

\int \frac{\mathrm{dx}}{x^2\sqrt{x^2\pm a^2}} = \mp \frac{\sqrt{x^2\pm a^2}}{a^2x} + C \tag{45}

\int \frac{\mathrm{dx}}{x^3\sqrt{x^2+a^2}} = -\frac{\sqrt{x^2+a^2}}{2a^2x^2} + \frac{1}{2a^3}\ln\left|\frac{a+\sqrt{x^2+a^2}}{x}\right| + C \tag{46}

\int \frac{\mathrm{dx}}{x^3\sqrt{x^2-a^2}} = \frac{\sqrt{x^2-a^2}}{2a^2x^2} + \frac{1}{2a^3}\arccos\frac{a}{|x|} + C \tag{47}

\int \frac{\sqrt{x^2+a^2}}{x} \, \mathrm{dx} = \sqrt{x^2+a^2} - a\ln\left|\frac{a+\sqrt{x^2+a^2}}{|x|}\right| + C \tag{48}

\int \frac{\sqrt{x^2-a^2}}{x} \, \mathrm{dx} = \sqrt{x^2-a^2} - a\arccos\frac{a}{|x|} + C \tag{49}

\int \frac{\sqrt{x^2\pm a^2}}{x^2} \, \mathrm{dx} = -\frac{\sqrt{x^2\pm a^2}}{x} + \ln\left|x+\sqrt{x^2\pm a^2}\right| + C \tag{50}

\int \frac{\mathrm{dx}}{x\sqrt{x^2+a^2}} = \frac{1}{a}\ln\left|\frac{x}{a+\sqrt{x^2+a^2}}\right| + C \tag{51}

\int \frac{\mathrm{dx}}{x\sqrt{x^2-a^2}} = \frac{1}{a}\arccos\frac{a}{|x|} + C \tag{52}

六、含有 $a+bx+cx^2$ 的积分

\int \frac{\mathrm{dx}}{a+bx+cx^2} = \begin{cases} \displaystyle \frac{2}{\sqrt{4ac-b^2}}\arctan\frac{2cx+b}{\sqrt{4ac-b^2}} + C, & \text{当 } b^2 < 4ac, \\ \displaystyle \frac{1}{\sqrt{b^2-4ac}}\ln\left|\frac{\sqrt{b^2-4ac}-b-2cx}{\sqrt{b^2-4ac}+b+2cx}\right| + C, & \text{当 } b^2 > 4ac. \end{cases} \tag{53}

七、含有 $\sqrt{a+bx+cx^2}$ 的积分

\int \frac{\mathrm{dx}}{\sqrt{a+bx+cx^2}} = \begin{cases} \displaystyle \frac{1}{\sqrt{c}}\ln\left|2cx+b+2\sqrt{c(a+bx+cx^2)}\right| + C, & \text{当 } c > 0, \\ \displaystyle \frac{1}{\sqrt{-c}}\arcsin\frac{2cx+b}{\sqrt{b^2-4ac}} + C, & \text{当 } c < 0, b^2 > 4ac. \end{cases} \tag{54}

\int \sqrt{a+bx+cx^2} \, \mathrm{dx} = \frac{2cx+b}{4c}\sqrt{a+bx+cx^2} + \frac{4ac-b^2}{8c}\int \frac{\mathrm{dx}}{\sqrt{a+bx+cx^2}} \tag{55}

\int \frac{x \, \mathrm{dx}}{\sqrt{a+bx+cx^2}} = \frac{1}{c}\sqrt{a+bx+cx^2} - \frac{b}{2c}\int \frac{\mathrm{dx}}{\sqrt{a+bx+cx^2}} \tag{56}

八、含有三角函数的积分

\int \sin^2 ax \, \mathrm{dx} = \frac{1}{2a}(ax - \sin ax\cos ax) + C \tag{57}

\int \cos^2 ax \, \mathrm{dx} = \frac{1}{2a}(ax + \sin ax\cos ax) + C \tag{58}

\int \sin^n x \, \mathrm{dx} = -\frac{1}{n}\sin^{n-1}x\cos x + \frac{n-1}{n}\int \sin^{n-2}x \, \mathrm{dx} \tag{59}

\int \cos^n x \, \mathrm{dx} = \frac{1}{n}\cos^{n-1}x\sin x + \frac{n-1}{n}\int \cos^{n-2}x \, \mathrm{dx} \tag{60}

\int \tan x \, \mathrm{dx} = -\ln|\cos x| + C \tag{61}

\int \cot x \, \mathrm{dx} = \ln|\sin x| + C \tag{62}

\int \tan^n x \, \mathrm{dx} = \frac{\tan^{n-1}x}{n-1} - \int \tan^{n-2}x \, \mathrm{dx} \tag{63}

\int \cot^n x \, \mathrm{dx} = -\frac{\cot^{n-1}x}{n-1} - \int \cot^{n-2}x \, \mathrm{dx} \tag{64}

\int \sec x \, \mathrm{dx} = \ln|\sec x + \tan x| + C \tag{65}

\int \csc x \, \mathrm{dx} = -\ln|\csc x + \cot x| + C \tag{66}

\int \sec^n x \, \mathrm{dx} = \frac{\tan x\sec^{n-2}x}{n-1} + \frac{n-2}{n-1}\int \sec^{n-2}x \, \mathrm{dx} \tag{67}

\int \csc^n x \, \mathrm{dx} = -\frac{\cot x\csc^{n-2}x}{n-1} + \frac{n-2}{n-1}\int \csc^{n-2}x \, \mathrm{dx} \tag{68}

\int \sec x\tan x \, \mathrm{dx} = \sec x + C \tag{69}

\int \csc x\cot x \, \mathrm{dx} = -\csc x + C \tag{70}

\int \sin ax\sin bx \, \mathrm{dx} = -\frac{\sin(a+b)x}{2(a+b)} + \frac{\sin(a-b)x}{2(a-b)} + C \quad (a^2 \neq b^2) \tag{71}

\int \sin ax\cos bx \, \mathrm{dx} = -\frac{\cos(a+b)x}{2(a+b)} - \frac{\cos(a-b)x}{2(a-b)} + C \quad (a^2 \neq b^2) \tag{72}

\int \cos ax\cos bx \, \mathrm{dx} = \frac{\sin(a+b)x}{2(a+b)} + \frac{\sin(a-b)x}{2(a-b)} + C \quad (a^2 \neq b^2) \tag{73}

\begin{aligned} \int \sin^m x\cos^n x \, \mathrm{dx} &= \frac{\sin^{m+1}x\cos^{n-1}x}{m+n} + \frac{n-1}{m+n}\int \sin^m x\cos^{n-2}x \, \mathrm{dx} \\ &= \frac{\sin^{m-1}x\cos^{n+1}x}{m+n} + \frac{m-1}{m+n}\int \sin^{m-2}x\cos^n x \, \mathrm{dx} \end{aligned} \tag{74}

\int \frac{\mathrm{dx}}{a+b\cos x} = \begin{cases} \displaystyle \frac{2}{\sqrt{a^2-b^2}}\arctan\left(\sqrt{\frac{a-b}{a+b}}\tan\frac{x}{2}\right) + C, & \text{当 } a^2 > b^2, \\ \displaystyle \frac{1}{\sqrt{b^2-a^2}}\ln\left|\frac{b+a\cos x+\sqrt{b^2-a^2}\sin x}{a+b\cos x}\right| + C, & \text{当 } a^2 < b^2. \end{cases} \tag{75}

\int e^{ax} \, \mathrm{dx} = e^{ax}(x-1) + C \tag{76}

\int x^n e^{ax} \, \mathrm{dx} = x^n e^{ax} - n\int x^{n-1}e^{ax} \, \mathrm{dx} \tag{77}

\int \ln x \, \mathrm{dx} = x(\ln x - 1) + C \tag{78}

\int x^n\ln x \, \mathrm{dx} = \frac{x^{n+1}}{(n+1)^2}[(n+1)\ln x - 1] + C \tag{79}

\int x\sin x \, \mathrm{dx} = \sin x - x\cos x + C \tag{80}

\int x\cos x \, \mathrm{dx} = \cos x + x\sin x + C \tag{81}

\int x^n\sin x \, \mathrm{dx} = -x^n\cos x + n\int x^{n-1}\cos x \, \mathrm{dx} \tag{82}

\int x^n\cos x \, \mathrm{dx} = x^n\sin x - n\int x^{n-1}\sin x \, \mathrm{dx} \tag{83}

\int e^{ax}\sin bx \, \mathrm{dx} = \frac{e^{ax}(a\sin bx - b\cos bx)}{a^2+b^2} + C \tag{84}

\int e^{ax}\cos bx \, \mathrm{dx} = \frac{e^{ax}(a\cos bx + b\sin bx)}{a^2+b^2} + C \tag{85}

\int \arcsin x \, \mathrm{dx} = x\arcsin x + \sqrt{1-x^2} + C \tag{86}

\int \arctan x \, \mathrm{dx} = x\arctan x - \frac{1}{2}\ln(1+x^2) + C \tag{87}

\int x^n\arcsin x \, \mathrm{dx} = \frac{1}{n+1}\left(x^{n+1}\arcsin x - \int \frac{x^{n+1}}{\sqrt{1-x^2}} \, \mathrm{dx}\right) \tag{88}

\int x^n\arctan x \, \mathrm{dx} = \frac{1}{n+1}\left(x^{n+1}\arctan x - \int \frac{x^{n+1}}{1+x^2} \, \mathrm{dx}\right) \tag{89}

附录 5 复数及其运算

一、复数的概念

二、复数的代数运算

三、复数的三种表示式及 Euler 公式

附录 6 简明积分表

一、含有 a+bxa+bx 的积分

二、含有 √(a+bx)a+bx​ 的积分

三、含有 a² ± x²a2±x2 的积分

四、含有 √(a²-x²)a2−x2​ 的积分

五、含有 √(x² ± a²)x2±a2​ 的积分

六、含有 a+bx+cx²a+bx+cx2 的积分

七、含有 √(a+bx+cx²)a+bx+cx2​ 的积分