附录

附录Ⅰ 二阶和三阶行列式简介

给出二元线性方程组

$$ \{\begin{aligned}a_{11}x_{1}+a_{12}x_{2}&=b_{1},\\ a_{21}x_{1}+a_{22}x_{2}&=b_{2},\end{aligned}. $$

求这方程组的解.

用大家熟知的消元法，分别消去方程组(1)中的 $ x_{2} $及 $ x_{1} $，得

$$ \{\begin{array}{l}(a_{11}a_{22}-a_{12}a_{21})x_{1}=b_{1}a_{22}-a_{12}b_{2},\\ (a_{11}a_{22}-a_{12}a_{21})x_{2}=a_{11}b_{2}-b_{1}a_{21}.\end{array}. $$

下面引入二阶行列式，然后利用二阶行列式来进一步讨论上述问题。设已知四个数排成正方形表

$$ \begin{pmatrix}a_{11}&a_{12}\\&\\a_{21}&a_{22}\end{pmatrix}, $$

则数 $ a_{11}a_{22}-a_{12}a_{21} $ 称为对应于这个表的二阶行列式，用记号

$$ \begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix} $$

表示，因此

$$ \left|\begin{array}{l l}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array}\right|=a_{11}a_{22}-a_{12}a_{21}. $$

数 $ a_{11}, a_{12}, a_{21}, a_{22} $ 叫做行列式(3)的元素，横排叫做 $ \uwave{\text{行}} $，竖排叫做 $ \uwave{\text{列}} $。元素 $ a_{ij} $ 中的第一个指标 i 和第二个指标 j 依次表示该元素所在的行数和列数。例如，元素 $ a_{21} $ 在行列式(3)中位于第二行和第一列。

现在，方程组(2)可利用行列式来表示. 设

$$ D=\left|\begin{array}{l l}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array}\right|=a_{11}a_{22}-a_{12}a_{21}, $$

$$ D_{1}=\left|\begin{array}{l l}b_{1}&a_{12}\\ b_{2}&a_{22}\end{array}\right|=b_{1}a_{22}-a_{12}b_{2}, $$

$$ D_{2}=\left|\begin{array}{l l}a_{11}&b_{1}\\ a_{21}&b_{2}\end{array}\right|=a_{11}b_{2}-b_{1}a_{21}, $$

则方程组(2)可写成

$$ \{\begin{aligned}D x_{1}&=D_{1},\\ D x_{2}&=D_{2}.\end{aligned}. $$

我们注意到，D 就是方程组(1)中 $ x_{1} $ 及 $ x_{2} $ 的系数构成的行列式，因此称为系数行列式，而 $ D_{1} $ 和 $ D_{2} $ 分别是用方程组(1)右端的常数项代替 D 的第一列和第二列而形成的.

若 $ D\neq0 $，则方程组（2）的解为

$$ x_{1}=\frac{D_{1}}{D},x_{2}=\frac{D_{2}}{D}. $$

把(4)中 $ x_{1} $ 及 $ x_{2} $ 的值代入方程组(1)，便可证实 $ x_{1} $ 及 $ x_{2} $ 的这对值也是方程组(1)的解. 另一方面，(2)是由(1)导出的，因此(1)的解一定是(2)的解. 现在(2)只有一组解(4)，所以(4)是方程组(1)的唯一解. 由此得出结论：

在 $ D\neq0 $的条件下，方程组(1)有唯一的解

$$ x_{1}=\frac{D_{1}}{D},x_{2}=\frac{D_{2}}{D}. $$

例1 解方程组

$$ \{\begin{aligned}&2x+3y=8,\\ &x-2y=-3.\end{aligned}. $$

解 $ D=\begin{vmatrix}2 & 3 \\ 1 & -2\end{vmatrix}=2\times(-2)-3\times1=-7 $,

$$ D_{1}=\left|\begin{matrix}8&3\\ -3&-2\end{matrix}\right|=8\times(-2)-3\times(-3)=-7, $$

$$ D_{2}=\left|\begin{matrix}2&8\\ 1&-3\end{matrix}\right|=2\times(-3)-8\times1=-14. $$

因 $ D = -7 \neq 0 $ ，故所给方程组有唯一解

$$ x=\frac{D_{1}}{D}=\frac{-7}{-7}=1,\quad y=\frac{D_{2}}{D}=\frac{-14}{-7}=2. $$

下面介绍三阶行列式概念.

设已知九个数排成正方形表

$$ \begin{pmatrix}a_{11}&a_{12}&\dot{a_{13}}\\&\\a_{21}&a_{22}&a_{23}\\&&\\a_{31}&a_{32}&a_{33}\end{pmatrix}, $$

则数 $ a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{12}a_{21}a_{33}-a_{11}a_{23}a_{32} $ 称为对应于这个表的三阶行列式，用记号

$$ \left|\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\&a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right| $$

表示，因此

$$ \begin{array}{l}\left|\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right|\\=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{12}a_{21}a_{33}-a_{11}a_{23}a_{32}.\end{array} $$

关于三阶行列式的元素、行、列等概念，与二阶行列式的相应概念类似，不再重复.

(5)式右端相当复杂,我们可以借助下列图形得出它的计算法则(通常称为对角线法则):

行列式中从左上角到右下角的直线称为 $ \uwave{\text{主对角线}} $，从右上角到左下角的直线称为 $ \uwave{\text{次对角线}} $。主对角线上元素的乘积以及位于主对角线的平行线上的元素与对角上的元素的乘积，前面都取正号。次对角线上元素的乘积以及位于次对角线的平行线上的元素与对角上的元素的乘积，前面都取负号。

$$ \begin{aligned}& 例 2\quad\left|\begin{matrix}{{{2}}}&{{{1}}}&{{{2}}} \\{{{-4}}}&{{{3}}}&{{{1}}} \\{{{2}}}&{{{3}}}&{{{5}}}\end{matrix}\right|\\&=2\times3\times5+1\times1\times2+2\times(-4)\times3-2\times3\times2-1\times(-4)\times5-2\times1\times3\\&=30+2-24-12+20-6=10.\end{aligned} $$

利用交换律及结合律，可把(5)式改写如下：

$$ \begin{array}{r l}&{\left|\begin{array}{l l l}{a_{11}}&{a_{12}}&{a_{13}}\\ {a_{21}}&{a_{22}}&{a_{23}}\\ {a_{31}}&{a_{32}}&{a_{33}}\end{array}\right|}\\ &{=a_{11}\left(\begin{array}{l l l}{a_{22}}&{a_{33}-a_{23}a_{32}}\\ {a_{33}}&{a_{32}}\end{array}\right)-a_{12}\left(\begin{array}{l l l}{a_{21}a_{33}-a_{23}a_{31}}\\ {a_{33}}\end{array}\right)+a_{13}\left(\begin{array}{l l l}{a_{21}a_{32}-a_{22}a_{31}}\\ {a_{31}}\end{array}\right).}\end{array} $$

把上式右端三个括号中的式子表示为二阶行列式，则有

$$ \begin{array}{|l|l|l|l|l|l|l|l|l|}\hline{a_{11}}&{a_{12}}&{a_{13}}&{...}&{...}&{}&{}&{...}&{...}\\ {a_{21}}&{a_{22}}&{a_{23}}&{...}&{a_{11}}&{\left|\begin{array}{l l}{a_{22}}&{a_{23}}\\ {...}&{...}\\ {a_{32}}&{a_{33}}\\ \end{array}\right|-a_{12}}&{\left|\begin{array}{l l}{a_{21}}&{a_{23}}\\ {...}&{...}\\ {a_{31}}&{a_{33}}\\ \end{array}\right|+a_{13}}&{\left|\begin{array}{l l}{a_{21}}&{a_{22}}\\ {...}&{...}\\ {a_{31}}&{a_{32}}\\ \end{array}\right|}.\end{array} $$

上式称为三阶行列式按第一行的展开式.

例3 将例2中的行列式按第一行展开并计算它的值.

解

$$ \begin{align*}\left|\begin{array}{ccc}{{{2}}}&{{{1}}}&{{{2}}} \\{{{-4}}}&{{{3}}}&{{{1}}} \\{{{2}}}&{{{3}}}&{{{5}}}\end{array}\right|&=2\left|\begin{array}{cc}{{{3}}}&{{{1}}} \\{{{3}}}&{{{5}}}\end{array}\right|-\left|\begin{array}{cc}{{{-4}}}&{{{1}}} \\{{{2}}}&{{{5}}}\end{array}\right|+2\left|\begin{array}{cc}{{{-4}}}&{{{3}}} \\{{{2}}}&{{{3}}}\end{array}\right|\\&=2\times12-(-22)+2\times(-18)\\&=24+22-36=10.\end{align*} $$

习题

1. 利用二阶行列式解下列方程组：

(1)

$$ \{\begin{aligned}5x-y&=2,\\ 3x+2y&=9;\end{aligned}. $$

(2)

$$ \{\begin{aligned}3x+4y&=2,\\ 2x+3y&=7.\end{aligned}. $$

1. 利用对角线法则，计算下列各行列式：

(1)

$$ \begin{vmatrix}2&0&1\\ 1&-4&-1\\ -1&8&3\end{vmatrix}; $$

(2)

$$ \left|\begin{array}{ccc}4&-2&4\\10&2&12\\1&2&2\end{array}\right|; $$

(3)

$$ \begin{vmatrix}3&4&2\\ 7&5&1\\ 3&2&4\end{vmatrix}; $$

(4)

$$ \left|\begin{array}{ccc}1&1&1\\1&1+a&1\\1&1&1+b\end{array}\right|. $$

1. 将下列行列式按第一行展开并计算它们的值：

(1)

$$ \begin{vmatrix}1&2&3\\ 3&1&2\\ 2&3&1\end{vmatrix}; $$

(2)

$$ \begin{vmatrix}{{{-1}}}&{{{2}}}&{{{2}}} \\{{{2}}}&{{{-1}}}&{{{2}}} \\{{{2}}}&{{{2}}}&{{{-1}}}\end{vmatrix}. $$

1. 证明下列等式：

(1)

$$ \begin{array}{|l|l|l|l|l|l|l|l|l|l}\hline{a_{11}}&{a_{12}}&{a_{13}}&{}\\ {a_{21}}&{a_{22}}&{a_{23}}&{}\\ {a_{31}}&{a_{32}}&{a_{33}}&{}\end{array}=-a_{21}\left|\begin{array}{l l|l}{a_{12}}&{a_{13}}&{}\\ {a_{32}}&{a_{33}}&{}\end{array}\right|+a_{22}\left|\begin{array}{l l|l}{a_{11}}&{a_{13}}&{}\\ {a_{31}}&{a_{33}}&{}\end{array}\right|-a_{23}\left|\begin{array}{l l|l}{a_{11}}&{a_{12}}&{}\\ {a_{31}}&{a_{32}}&{}\end{array}\right|; $$

(2)

$$ \begin{array}{|l|l|l|l|l|l|l|l|l|}\hline{a_{11}}&{a_{12}}&{a_{13}}&{}&{}&{}&{}&{}\\ \hline{a_{21}}&{a_{22}}&{a_{23}}&{}\\ {}&{}&{}&{a_{31}}&{a_{12}}&{a_{13}}&{a_{32}}&{a_{11}}&{a_{13}}\\ \hline{a_{31}}&{a_{32}}&{a_{33}}&{}&{}&{a_{22}}&{a_{23}}&{a_{21}}&{a_{23}}\\ \end{array}, $$

注：上面这两个等式分别称为三阶行列式按第二行和按第三行的展开式.

答案

1. (1) x = 1, y = 3; (2) x = -22, y = 17.

1. (1) -4; (2) 8; (3) -48; (4) ab.

3.（1）18；（2）27.

1. 略.

附录Ⅱ 基本初等函数的图形

$$ y=x^{\mu} $$

反三角函数

附录Ⅲ 几种常用的曲线

（1）三次抛物线

（2）半立方抛物线

$$ y=ax^{3} $$

（3）概率曲线

$$ y=\mathrm{e}^{-x^{2}} $$

$$ y^{2}=ax^{3} $$

（4）箕舌线

$$ y^{2}(2a-x)=x^{3} $$

$$ y=\frac{8a^{3}}{x^{2}+4a^{2}} $$

（5）蔓叶线

（6）笛卡儿叶形线

$$ x^{3}+y^{3}-3axy=0 $$

$$ x=\frac{3at}{1+t^{3}},y=\frac{3at^{2}}{1+t^{3}} $$

（7）星形线（内摆线的一种）

（8） 摆线

$$ x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} $$

$$ \{\begin{aligned}x&=a\cos^{3}\theta\\ y&=a\sin^{3}\theta\end{aligned}. $$

$$ \{\begin{aligned}x&=a(\theta-\sin\theta)\\ y&=a(1-\cos\theta)\end{aligned}. $$

(9) 心形线（外摆线的一种）

（10）阿基米德螺线

$$ x^{2}+y^{2}+ax=a\sqrt{x^{2}+y^{2}} $$

$$ \rho=a\left(1-\cos\ \theta\right) $$

（11）对数螺线

（12）双曲螺线

（13）伯努利双纽线

（14）伯努利双纽线

$$ \rho^{2}=a^{2}\sin2\theta $$

$$ (x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2}) $$

$$ \rho^{2}=a^{2}\cos2\theta $$

（15）三叶玫瑰线

（16）三叶玫瑰线

（17）四叶玫瑰线

（18）四叶玫瑰线

$ \rho = a \sin 2\theta $

$ \rho = a \cos 2\theta $

$$ \begin{aligned}&10.\ \int\sqrt{ax+b}\mathrm{d}x=\frac{2}{3a}\sqrt{(ax+b)^{3}}+C.\\ &11.\ \int x\sqrt{ax+b}\mathrm{d}x=\frac{2}{15a^{2}}(3ax-2b)\sqrt{(ax+b)^{3}}+C.\\ &12.\ \int x^{2}\sqrt{ax+b}\mathrm{d}x=\frac{2}{105a^{3}}(15a^{2}x^{2}-12abx+8b^{2})\sqrt{(ax+b)^{3}}+C.\\ &13.\ \int\frac{x}{\sqrt{ax+b}}\mathrm{d}x=\frac{2}{3a^{2}}(ax-2b)\sqrt{ax+b}+C.\\ &14.\ \int\frac{x^{2}}{\sqrt{ax+b}}\mathrm{d}x=\frac{2}{15a^{3}}(3a^{2}x^{2}-4abx+8b^{2})\sqrt{ax+b}+C.\\ \end{aligned} $$

（二）含有 $ \sqrt{ax+b} $的积分

$$ \begin{aligned}5.&\int\frac{\mathrm{d}x}{x\left(ax+b\right)}=-\frac{1}{b}\ln\left|\frac{ax+b}{x}\right|+C.\\6.&\int\frac{\mathrm{d}x}{x^{2}\left(ax+b\right)}=-\frac{1}{bx}+\frac{a}{b^{2}}\ln\left|\frac{ax+b}{x}\right|+C.\\7.&\int\frac{x}{\left(ax+b\right)^{2}}\mathrm{d}x=\frac{1}{a^{2}}\bigg(\ln\left|ax+b\right|+\frac{b}{ax+b}\bigg)+C.\\8.&\int\frac{x^{2}}{\left(ax+b\right)^{2}}\mathrm{d}x=\frac{1}{a^{3}}\bigg(ax+b-2b\ln\left|ax+b\right|-\frac{b^{2}}{ax+b}\bigg)+C.\\9.&\int\frac{\mathrm{d}x}{x\left(ax+b\right)^{2}}=\frac{1}{b\left(ax+b\right)}-\frac{1}{b^{2}}\ln\left|\frac{ax+b}{x}\right|+C.\end{aligned} $$

$$ \begin{align*}&1.\ \int\frac{\mathrm{d}x}{ax+b}=\frac{1}{a}\ln|ax+b|+C.\\&2.\ \int(ax+b)^{\mu}\mathrm{d}x=\frac{1}{a(\mu+1)}(ax+b)^{\mu+1}+C(\mu\neq-1).\\&3.\ \int\frac{x}{ax+b}\mathrm{d}x=\frac{1}{a^{2}}(ax+b-b\ln|ax+b|)+C.\\&4.\ \int\frac{x^{2}}{ax+b}\mathrm{d}x=\frac{1}{a^{3}}\left[\frac{1}{2}(ax+b)^{2}-2b(ax+b)+b^{2}\ln|ax+b|\right]+C.\end{align*} $$

（一）含有 $ ax+b $的积分

公众号：考研讲课

附录IV 积分表

$$ \begin{aligned}&23.\ \int\frac{x}{ax^{2}+b}\mathrm{d}x=\frac{1}{2a}\ln|ax^{2}+b|+C.\\ &24.\ \int\frac{x^{2}}{ax^{2}+b}\mathrm{d}x=\frac{x}{a}-\frac{b}{a}\int\frac{\mathrm{d}x}{ax^{2}+b}.\\ &25.\ \int\frac{\mathrm{d}x}{x(ax^{2}+b)}=\frac{1}{2b}\ln\frac{x^{2}}{|ax^{2}+b|}+C.\\ &26.\ \int\frac{\mathrm{d}x}{x^{2}(ax^{2}+b)}=-\frac{1}{bx}-\frac{a}{b}\int\frac{\mathrm{d}x}{ax^{2}+b}.\\ &27.\ \int\frac{\mathrm{d}x}{x^{3}(ax^{2}+b)}=\frac{a}{2b^{2}}\ln\frac{|ax^{2}+b|}{x^{2}}-\frac{1}{2bx^{2}}+C.\\ \end{aligned} $$

$$ 22.\int\frac{\mathrm{d}x}{ax^{2}+b}=\{\begin{matrix}\frac{1}{\sqrt{ab}}\arctan\sqrt{\frac{a}{b}}x+C&(b>0)\end{matrix}.,\\\{\frac{1}{2\sqrt{-ab}}\ln|\frac{\sqrt{a}x-\sqrt{-b}}{\sqrt{a}x+\sqrt{-b}}|.+C&(b<0).\end{matrix}. $$

（四）含有 $ ax^{2}+b $ (a>0) 的积分

$$ \begin{align*}19.\int\frac{\mathrm{d}x}{x^{2}+a^{2}}&=\frac{1}{a}\arctan\frac{x}{a}+C.\\20.\int\frac{\mathrm{d}x}{\left(x^{2}+a^{2}\right)^{n}}&=\frac{x}{2\left(n-1\right)a^{2}\left(x^{2}+a^{2}\right)^{n-1}}+\frac{2n-3}{2\left(n-1\right)a^{2}}\int\frac{\mathrm{d}x}{\left(x^{2}+a^{2}\right)^{n-1}},\\21.\int\frac{\mathrm{d}x}{x^{2}-a^{2}}&=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C.\end{align*} $$

（三）含有 $ x^{2}\pm a^{2} $的积分

$$ \begin{aligned}&15.\ \int\frac{\mathrm{d}x}{x\sqrt{ax+b}}=\{\begin{aligned}&\frac{1}{\sqrt{b}}\ln|\frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}}|+C&(b>0)\\&\frac{2}{\sqrt{-b}}\arctan\sqrt{\frac{ax+b}{-b}}+C&(b<0)\end{aligned}.\\ &16.\ \int\frac{\mathrm{d}x}{x^{2}\sqrt{ax+b}}=-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}\int\frac{\mathrm{d}x}{x\sqrt{ax+b}}.\\ &17.\ \int\frac{\sqrt{ax+b}}{x}\mathrm{d}x=2\sqrt{ax+b}+b\int\frac{\mathrm{d}x}{x\sqrt{ax+b}}.\\ &18.\ \int\frac{\sqrt{ax+b}}{x^{2}}\mathrm{d}x=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int\frac{\mathrm{d}x}{x\sqrt{ax+b}}.\\ \end{aligned} $$

$$ \begin{aligned}&31.\ \int\frac{\mathrm{d}x}{\sqrt{x^{2}+a^{2}}}=\operatorname{arsh}\frac{x}{a}+C_{1}=\ln\left(x+\sqrt{x^{2}+a^{2}}\right)+C.\\ &\\ &32.\ \int\frac{\mathrm{d}x}{\sqrt{\left(x^{2}+a^{2}\right)^{3}}}=\frac{x}{a^{2}\sqrt{x^{2}+a^{2}}}+C.\\ &\\ &33.\ \int\frac{x}{\sqrt{x^{2}+a^{2}}}\mathrm{d}x=\sqrt{x^{2}+a^{2}}+C.\\ &\\ &34.\ \int\frac{x}{\sqrt{\left(x^{2}+a^{2}\right)^{3}}}\mathrm{d}x=-\frac{1}{\sqrt{x^{2}+a^{2}}}+C.\\ &\\ &35.\ \int\frac{x^{2}}{\sqrt{x^{2}+a^{2}}}\mathrm{d}x=\frac{x}{2}\sqrt{x^{2}+a^{2}}-\frac{a^{2}}{2}\ln\left(x+\sqrt{x^{2}+a^{2}}\right)+C.\\ &\\ &36.\ \int\frac{x^{2}}{\sqrt{\left(x^{2}+a^{2}\right)^{3}}}\mathrm{d}x=-\frac{x}{\sqrt{x^{2}+a^{2}}}+\ln\left(x+\sqrt{x^{2}+a^{2}}\right)+C.\\ &\\ &37.\ \int\frac{\mathrm{d}x}{x\sqrt{x^{2}+a^{2}}}=\frac{1}{a}\ln\frac{\sqrt{x^{2}+a^{2}}-a}{|x|}+C.\\ &\\ &38.\ \int\frac{\mathrm{d}x}{x^{2}\sqrt{x^{2}+a^{2}}}=-\frac{\sqrt{x^{2}+a^{2}}}{a^{2}x}+C.\\ &\\ &39.\ \int\sqrt{x^{2}+a^{2}}\mathrm{d}x=\frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}\ln\left(x+\sqrt{x^{2}+a^{2}}\right)+C.\\ &\\ &40.\ \int\sqrt{\left(x^{2}+a^{2}\right)^{3}}\mathrm{d}x=\frac{x}{8}\left(2x^{2}+5a^{2}\right)\sqrt{x^{2}+a^{2}}+\frac{3}{8}a^{4}\ln\left(x+\sqrt{x^{2}+a^{2}}\right)+C.\\ &\\ &41.\ \int x\sqrt{x^{2}+a^{2}}\mathrm{d}x=\frac{1}{3}\sqrt{\left(x^{2}+a^{2}\right)^{3}}+C.\\ \end{aligned} $$

（六）含有 $ \sqrt{x^{2}+a^{2}} $（a>0）的积分

$$ \begin{aligned}&29.\ \int\frac{\mathrm{d}x}{ax^{2}+bx+c}=\{\begin{aligned}&\frac{2}{\sqrt{4ac-b^{2}}}\arctan\frac{2ax+b}{\sqrt{4ac-b^{2}}}+C\quad(b^{2}<4ac),\\&\frac{1}{\sqrt{b^{2}-4ac}}\ln|\frac{2ax+b-\sqrt{b^{2}-4ac}}{2ax+b+\sqrt{b^{2}-4ac}}|+C\quad(b^{2}>4ac).\end{aligned}.\\&30.\ \int\frac{x}{ax^{2}+bx+c}\mathrm{d}x=\frac{1}{2a}\ln|ax^{2}+bx+c|-\frac{b}{2a}\int\frac{\mathrm{d}x}{ax^{2}+bx+c}.\end{aligned} $$

（五）含有 $ ax^{2}+bx+c $ (a>0) 的积分

$$ 28.\int\frac{\mathrm{d}x}{\left(ax^{2}+b\right)^{2}}=\frac{x}{2b\left(ax^{2}+b\right)}+\frac{1}{2b}\int\frac{\mathrm{d}x}{ax^{2}+b}. $$

$$ \int\frac{\sqrt{x^{2}-a^{2}}}{x}dx=\sqrt{x^{2}-a^{2}}-a\arccos\frac{a}{|x|}+C. $$

$$ \begin{array}{l}55.\ \int x\sqrt{x^{2}-a^{2}}\mathrm{d}x=\frac{1}{3}\sqrt{\left(x^{2}-a^{2}\right)^{3}}+C.\\56.\ \int x^{2}\sqrt{x^{2}-a^{2}}\mathrm{d}x=\frac{x}{8}\left(2x^{2}-a^{2}\right)\sqrt{x^{2}-a^{2}}-\frac{a^{4}}{8}\ln|x+\sqrt{x^{2}-a^{2}}|+C.\end{array} $$

$$ \begin{array}{l}52.\ \displaystyle\int\frac{\mathrm{d}x}{x^{2}\sqrt{x^{2}-a^{2}}}=\frac{\sqrt{x^{2}-a^{2}}}{a^{2}x}+C.\\53.\ \displaystyle\int\frac{x}{\sqrt{x^{2}-a^{2}}}\mathrm{d}x=\frac{x}{2}\sqrt{\frac{x^{2}-a^{2}}{x^{2}-a^{2}}}-\frac{a^{2}}{2}\mathrm{ln}|x+\sqrt{x^{2}-a^{2}}|+C.\\54.\ \displaystyle\int\frac{x}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}}\mathrm{d}x=\frac{x}{8}\left(2x^{2}-5a^{2}\right)\sqrt{x^{2}-a^{2}}+\frac{3}{8}a^{4}\ln|x+\sqrt{x^{2}-a^{2}}|+C.\\\end{array} $$

$$ \int\frac{\mathrm{d}x}{x\sqrt{x^{2}-a^{2}}}=\frac{1}{a}\arccos\frac{a}{|x|}+C. $$

$$ \int\frac{x^{2}}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}}\mathrm{d}x=-\frac{x}{\sqrt{x^{2}-a^{2}}}+\ln|x+\sqrt{x^{2}-a^{2}}|+C. $$

$$ 48.\int\frac{x}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}}\mathrm{d}x=-\frac{1}{\sqrt{x^{2}-a^{2}}}+C. $$

$$ 47.\int\frac{x}{\sqrt{x^{2}-a^{2}}}dx=\sqrt{x^{2}-a^{2}}+C. $$

$$ 46.\int\frac{\mathrm{d}x}{\sqrt{\left(x^{2}-a^{2}\right)^{3}}}=-\frac{x}{a^{2}\sqrt{x^{2}-a^{2}}}+C. $$

$$ \int\frac{\mathrm{d}x}{\sqrt{x^{2}-a^{2}}}=\frac{x}{\left|x\right|}\operatorname{arch}\frac{\left|x\right|}{a}+C_{1}=\ln\left|x+\sqrt{x^{2}-a^{2}}\right|+C. $$

（七）含有 $ \sqrt{x^{2}-a^{2}} $（a>0）的积分

$$ \begin{array}{r l}&{43.\quad\displaystyle\int\frac{\sqrt{x^{2}+a^{2}}}{x}\mathrm{d}x=\sqrt{x^{2}+a^{2}}+a\ln\frac{\sqrt{x^{2}+a^{2}}-a}{\mid x\mid}+C.}\\ &{\quad44.\quad\displaystyle\int\frac{\sqrt{x^{2}+a^{2}}}{x^{2}}\mathrm{d}x=-\frac{\sqrt{x^{2}+a^{2}}}{x}+\ln\left(x+\sqrt{x^{2}+a^{2}}\right)+C.}\end{array} $$

$$ \int x^{2}\sqrt{x^{2}+a^{2}}\mathrm{d}x=\frac{x}{8}(2x^{2}+a^{2})\sqrt{x^{2}+a^{2}}-\frac{a^{4}}{8}\ln(x+\sqrt{x^{2}+a^{2}})+C. $$

公众号：考研讲课

58.

$$ \int\frac{\sqrt{x^{2}-a^{2}}}{x^{2}}\mathrm{d}x=-\frac{\sqrt{x^{2}-a^{2}}}{x}+\ln|x+\sqrt{x^{2}-a^{2}}|+C. $$

（八）含有 $ \sqrt{a^{2}-x^{2}} $（a>0）的积分

59.

$$ \int\frac{\mathrm{d}x}{\sqrt{a^{2}-x^{2}}}=\arcsin\frac{x}{a}+C. $$

60.

$$ \int\frac{\mathrm{d}x}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}}=\frac{x}{a^{2}\sqrt{a^{2}-x^{2}}}+C. $$

61.

$$ \int\frac{x}{\sqrt{a^{2}-x^{2}}}\mathrm{d}x=-\sqrt{a^{2}-x^{2}}+C. $$

62.

$$ \int\frac{x}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}}\mathrm{d}x=\frac{1}{\sqrt{a^{2}-x^{2}}}+C. $$

63.

$$ \int\frac{x^{2}}{\sqrt{a^{2}-x^{2}}}\mathrm{d}x=-\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}\arcsin\frac{x}{a}+C. $$

64.

$$ \int\frac{x^{2}}{\sqrt{\left(a^{2}-x^{2}\right)^{3}}}\mathrm{d}x=\frac{x}{\sqrt{a^{2}-x^{2}}}-\arcsin\frac{x}{a}+C. $$

65.

$$ \int\frac{\mathrm{d}x}{x\sqrt{a^{2}-x^{2}}}=\frac{1}{a}\ln\frac{a-\sqrt{a^{2}-x^{2}}}{|x|}+C. $$

66.

$$ \int\frac{\mathrm{d}x}{x^{2}\sqrt{a^{2}-x^{2}}}=-\frac{\sqrt{a^{2}-x^{2}}}{a^{2}x}+C. $$

67.

$$ \int\sqrt{a^{2}-x^{2}}\mathrm{d}x=\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}\arcsin\frac{x}{a}+C. $$

68.

$$ \int\sqrt{\left(a^{2}-x^{2}\right)^{3}}\mathrm{d}x=\frac{x}{8}\left(5a^{2}-2x^{2}\right)\sqrt{a^{2}-x^{2}}+\frac{3}{8}a^{4}\arcsin\frac{x}{a}+C. $$

69.

$$ \int x\sqrt{a^{2}-x^{2}}\mathrm{d}x=-\frac{1}{3}\sqrt{\left(a^{2}-x^{2}\right)^{3}}+C. $$

70.

$$ \int x^{2}\sqrt{a^{2}-x^{2}}\mathrm{d}x=\frac{x}{8}(2x^{2}-a^{2})\sqrt{a^{2}-x^{2}}+\frac{a^{4}}{8}\arcsin\frac{x}{a}+C. $$

71.

$$ \int\frac{\sqrt{a^{2}-x^{2}}}{x}\mathrm{d}x=\sqrt{a^{2}-x^{2}}+a\ln\frac{a-\sqrt{a^{2}-x^{2}}}{|x|}+C. $$

$$ 72.\ \int\frac{\sqrt{a^{2}-x^{2}}}{x^{2}}\mathrm{d}x=-\frac{\sqrt{a^{2}-x^{2}}}{x}-\arcsin\frac{x}{a}+C. $$

$$ \begin{aligned}81.\int\frac{\mathrm{d}x}{\sqrt{\left(x-a\right)\left(b-x\right)}}&=2\arcsin\sqrt[x-a]{\frac{b-a}{b-a}}+C(a

$$ \int\sqrt{\frac{x-a}{b-x}}\mathrm{d}x=(x-b)\sqrt{\frac{x-a}{b-x}}+(b-a)\arcsin\sqrt{\frac{x-a}{b-a}}+C. $$

$$ \int\sqrt{\frac{x-a}{x-b}}\mathrm{d}x=(x-b)\sqrt{\frac{x-a}{x-b}}+(b-a)\ln(\sqrt{|x-a|}+\sqrt{|x-b|})+C. $$

（十）含有 $ \sqrt{\pm\frac{x-a}{x-b}} $或 $ \sqrt{(x-a)(b-x)} $的积分

$$ \begin{aligned}&76.\ \int\frac{\mathrm{d}x}{\sqrt{c+bx-ax^{2}}}=\frac{1}{\sqrt{a}}\arcsin\frac{2ax-b}{\sqrt{b^{2}+4ac}}+C.\\ &77.\ \int\sqrt{c+bx-ax^{2}}\mathrm{d}x=\frac{2ax-b}{4a}\sqrt{c+bx-ax^{2}}+\\ &\quad\frac{b^{2}+4ac}{8\sqrt{a^{3}}}\arcsin\frac{2ax-b}{\sqrt{b^{2}+4ac}}+C.\\ &78.\ \int\frac{x}{\sqrt{c+bx-ax^{2}}}\mathrm{d}x=-\frac{1}{a}\sqrt{c+bx-ax^{2}}+\frac{b}{2\sqrt{a^{3}}}\arcsin\frac{2ax-b}{\sqrt{b^{2}+4ac}}+C.\\ \end{aligned} $$

$$ \begin{aligned}75.\ \int\frac{x}{\sqrt{ax^{2}+bx+c}}\mathrm{d}x&=\frac{1}{a}\sqrt{ax^{2}+bx+c}-\\&\quad\frac{b}{2\sqrt{a^{3}}}\ln|2ax+b+2\sqrt{a}\sqrt{ax^{2}+bx+c}|+C.\end{aligned} $$

$$ \begin{array}{c}74.\ \int\sqrt{ax^{2}+bx+c}\mathrm{d}x=\frac{2ax+b}{4a}\sqrt{ax^{2}+bx+c}+\\ \frac{4ac-b^{2}}{8\sqrt{a^{3}}}\ln\left|2ax+b+2\sqrt{a}\sqrt{ax^{2}+bx+c}\right|+C.\end{array} $$

$$ \int\frac{\mathrm{d}x}{\sqrt{ax^{2}+bx+c}}=\frac{1}{\sqrt{a}}\ln|2ax+b+2\sqrt{a}\sqrt{ax^{2}+bx+c}|+C. $$

（九）含有 $ \sqrt{\pm ax^{2}+bx+c} $（a>0）的积分

$$ \int\sin a x\sin b x d x=-\frac{1}{2(a+b)}\sin(a+b)x+\frac{1}{2(a-b)}\sin(a-b)x+C. $$

$$ \int\sin a x\cos b x d x=-\frac{1}{2(a+b)}\cos(a+b)x-\frac{1}{2(a-b)}\cos(a-b)x+C. $$

$$ =-\frac{1}{m+n}\cos^{m+1}x\sin^{n-1}x+\frac{n-1}{m+n}\int\cos^{m}x\sin^{n-2}x\mathrm{d}x. $$

$$ 99.\quad\int\cos^{m}x\sin^{n}x\mathrm{d}x=\frac{1}{m+n}\cos^{m-1}x\sin^{n+1}x+\frac{m-1}{m+n}\int\cos^{m-2}x\sin^{n}x\mathrm{d}x $$

$$ \int\frac{\mathrm{d}x}{\cos^{n}x}=\frac{1}{n-1}\cdot\frac{\sin x}{\cos^{n-1}x}+\frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\cos^{n-2}x} $$

$$ 97.\quad\int\frac{\mathrm{d}x}{\sin^{n}x}=-\frac{1}{n-1}\cdot\frac{\cos x}{\sin^{n-1}x}+\frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\sin^{n-2}x}. $$

$$ 96.\quad\int\cos^{n}x\mathrm{d}x=\frac{1}{n}\cos^{n-1}x\sin x+\frac{n-1}{n}\int\cos^{n-2}x\mathrm{d}x. $$

$$ \begin{aligned}&93.\ \int\sin^{2}x\mathrm{d}x=\frac{x}{2}-\frac{1}{4}\sin2x+C.\\ &\\ &94.\ \int\cos^{2}x\mathrm{d}x=\frac{x}{2}+\frac{1}{4}\sin2x+C.\\ &\\ &95.\ \int\sin^{n}x\mathrm{d}x=-\frac{1}{n}\sin^{n-1}x\cos x+\frac{n-1}{n}\int\sin^{n-2}x\mathrm{d}x.\\ \end{aligned} $$

1. $ \int \csc x \cot x \, dx = -\csc x + C $.

1. $ \int \sec x \tan x \, dx = \sec x + C $.

$$ 90.\int\csc^{2}x\mathrm{d}x=-\cot x+C. $$

1. $ \int \sec^2 x \, dx = \tan x + C $.

$$ 88.\ \int\csc\ x\mathrm{d}x=\ln\left|\tan\ \frac{x}{2}\right|+C=\ln|\csc\ x-\cot\ x|+C. $$

$$ 87.\quad\int\sec x\mathrm{d}x=\ln\left|\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\right|+C=\ln\ln\sec x+\tan x\ln+C. $$

1. $ \int \cot x \, dx = \ln |\sin x| + C $.

$$ 85.\int\tan x\mathrm{d}x=-\ln|\cos x|+C. $$

公众号：考研讲课

$$ 84.\int\cos x\mathrm{d}x=\sin x+C. $$

$$ \begin{aligned}&113.\ \int\arcsin\frac{x}{a}\mathrm{d}x=x\arcsin\frac{x}{a}+\sqrt{a^{2}-x^{2}}+C.\\ &114.\ \int x\arcsin\frac{x}{a}\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{a^{2}}{4}\right)\arcsin\frac{x}{a}+\frac{x}{4}\sqrt{a^{2}-x^{2}}+C.\\ &115.\ \int x^{2}\arcsin\frac{x}{a}\mathrm{d}x=\frac{x^{3}}{3}\arcsin\frac{x}{a}+\frac{1}{9}\left(x^{2}+2a^{2}\right)\sqrt{a^{2}-x^{2}}+C.\\ &116.\ \int\arccos\frac{x}{a}\mathrm{d}x=x\arccos\frac{x}{a}-\sqrt{a^{2}-x^{2}}+C.\\ \end{aligned} $$

（十二）含有反三角函数的积分（其中 a>0）

$$ \begin{aligned}&102.\int\cos ax\cos bx d x=\frac{2}{2\left(a+b\right)}\frac{\left(\sin\frac{x}{2}+\frac{1}{2\left(a-b\right)}\right)}{\left(\sin\frac{x}{2}+\frac{1}{2\left(a-b\right)}\right)}\sin\left(a-b\right)x+\\ &\begin{aligned}\\ &103.\int\frac{\mathrm{d}x}{a+b\sin x}=\frac{2}{\sqrt{a^{2}-b^{2}}}\arctan\frac{a\tan\frac{x}{2}+b}{\sqrt{a^{2}-b^{2}}}+C\left(a^{2}>b^{2}\right).\\&104.\int\frac{\mathrm{d}x}{a+b\sin x}=\frac{1}{\sqrt{b^{2}-a^{2}}}\ln\left|\frac{a\tan\frac{x}{2}+b-\sqrt{b^{2}-a^{2}}}{\arctan\frac{x}{2}+b+\sqrt{b^{2}-a^{2}}}\right|+C\left(a^{2}b^{2}\right).\\&106.\int\frac{\mathrm{d}x}{a+b\cos x}=\frac{1}{a+b}\sqrt{\frac{a+b}{b-a}}\ln\left|\frac{\tan\frac{x}{2}+\sqrt{\frac{a+b}{b-a}}}{\tan\frac{x}{2}-\sqrt{\frac{a+b}{b-a}}}\right|+C\left(a^{2}

$$ \begin{aligned}&122.\ \int a^{x}\mathrm{d}x=\frac{1}{\ln a}a^{x}+C.\\ &123.\ \int\mathrm{e}^{ax}\mathrm{d}x=-\frac{1}{a}\mathrm{e}^{ax}+C.\\ &124.\ \int x\mathrm{e}^{ax}\mathrm{d}x=\frac{1}{a^{2}}(ax-1)\mathrm{e}^{ax}+C.\\ &125.\ \int x^{n}\mathrm{e}^{ax}\mathrm{d}x=\frac{1}{a}x^{n}\mathrm{e}^{ax}-\frac{n}{a}\int x^{n-1}\mathrm{e}^{ax}\mathrm{d}x.\\ &126.\ \int x a^{x}\mathrm{d}x=\frac{x}{\ln a}a^{x}-\frac{1}{(\ln a)^{2}}a^{x}+C.\\ &127.\ \int x^{n}a^{x}\mathrm{d}x=\frac{1}{\ln a}x^{n}a^{x}-\frac{n}{\ln a}\int x^{n-1}a^{x}\mathrm{d}x.\\ &128.\ \int\mathrm{e}^{ax}\sin b x\mathrm{d}x=\frac{1}{a^{2}+b^{2}}\mathrm{e}^{ax}(asin bx-b\cos bx)+C.\\ &129.\ \int\mathrm{e}^{ax}\cos b x\mathrm{d}x=\frac{1}{a^{2}+b^{2}}\mathrm{e}^{ax}(bsin bx+acos bx)+C.\\ &130.\ \int\mathrm{e}^{ax}\sin^{n}b x\mathrm{d}x=\frac{1}{a^{2}+b^{2}n^{2}}\mathrm{e}^{ax}\sin^{n-1}bx(asin bx-nbcos bx)+\\ &\quad\frac{n(n-1)}{a^{2}+b^{2}n^{2}}\int\mathrm{e}^{ax}\sin^{n-2}b x\mathrm{d}x.\\ &131.\ \int\mathrm{e}^{ax}\cos^{n}b x\mathrm{d}x=\frac{1}{a^{2}+b^{2}n^{2}}\mathrm{e}^{ax}\cos^{n-1}bx(acos bx+nbsin bx)+\\ &\quad\frac{n(n-1)}{a^{2}+b^{2}n^{2}}\int\mathrm{e}^{ax}\cos^{n-2}b x\mathrm{d}x.\\ \end{aligned} $$

（十三） 含有指数函数的积分

$$ \begin{aligned}&117.\int x\arccos\frac{x}{a}\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{a^{2}}{4}\right)\arccos\frac{x}{a}-\frac{x}{4}\sqrt{a^{2}-x^{2}}+C.\\&118.\int x^{2}\arccos\frac{x}{a}\mathrm{d}x=\frac{x^{3}}{3}\arccos\frac{x}{a}-\frac{1}{9}\left(x^{2}+2a^{2}\right)\sqrt{a^{2}-x^{2}}+C.\\&119.\int\arctan\frac{x}{a}\mathrm{d}x=x\arctan\frac{x}{a}-\frac{a}{2}\ln\left(a^{2}+x^{2}\right)+C.\\&120.\int x\arctan\frac{x}{a}\mathrm{d}x=\frac{1}{2}\left(a^{2}+x^{2}\right)\arctan\frac{x}{a}-\frac{a}{2}x+C.\\&121.\int x^{2}\arctan\frac{x}{a}\mathrm{d}x=\frac{x^{3}}{3}\arctan\frac{x}{a}-\frac{a}{6}x^{2}+\frac{a^{3}}{6}\ln\left(a^{2}+x^{2}\right)+C.\\ \end{aligned} $$

（十四） 含有对数函数的积分

132.

$$ \int\ln x\mathrm{d}x=x\ln x-x+C. $$

133.

$$ \int\frac{\mathrm{d}x}{x\ln x}=\ln|\ln x|+C. $$

134.

$$ \int x^{n}\ln x\mathrm{d}x=\frac{1}{n+1}x^{n+1}\left(\ln x-\frac{1}{n+1}\right)+C. $$

135.

$$ \int\left(\ln x\right)^{n}\mathrm{d}x=x\left(\ln x\right)^{n}-n\int\left(\ln x\right)^{n-1}\mathrm{d}x. $$

136.

$$ \int x^{m}\left(\ln x\right)^{n}\mathrm{d}x=\frac{1}{m+1}x^{m+1}\left(\ln x\right)^{n}-\frac{n}{m+1}\int x^{m}\left(\ln x\right)^{n-1}\mathrm{d}x. $$

（十五） 含有双曲函数的积分

137.

$$ \int\mathrm{sh}x\mathrm{d}x=\mathrm{ch}x+C. $$

138.

$$ \int\mathrm{ch}x\mathrm{d}x=\mathrm{sh}x+C. $$

139.

$$ \int\mathrm{th}x\mathrm{d}x=\ln\mathrm{ch}x+C. $$

140.

$$ \int\mathrm{sh}^{2}x\mathrm{d}x=-\frac{x}{2}+\frac{1}{4}\mathrm{sh}2x+C. $$

141.

$$ \int\mathrm{ch}^{2}x\mathrm{d}x=\frac{x}{2}+\frac{1}{4}\mathrm{sh}2x+C. $$

（十六） 定积分

142.

$$ \int_{-\pi}^{\pi}\cos nx dx=\int_{-\pi}^{\pi}\sin nx dx=0. $$

143.

$$ \int_{-\pi}^{\pi}\cos m x\sin n x\mathrm{d}x=0. $$

144.

$$ \int_{-\pi}^{\pi}\cos m x\cos n x\mathrm{d}x=\{\begin{aligned}&0,&m\neq n,\\ &\pi,&m=n.\end{aligned}. $$

145.

$$ \int_{-\pi}^{\pi}\sin m x\sin n x\mathrm{d}x=\{\begin{array}{l}0,\quad m\neq n,\\ \pi,\quad m=n.\end{array}. $$

146.

$$ \int_{0}^{\pi}\sin m x\sin n x\mathrm{d}x=\int_{0}^{\pi}\cos m x\cos n x\mathrm{d}x=\{\frac{\pi}{2},\quad m\neq n\} $$

$$ \{\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\cdots\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}\quad(n\text{ 为正偶数})..,I_{0}=\frac{\pi}{2}.. $$

$$ \int\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\cdots\cdot\frac{4}{5}\cdot\frac{2}{3}\ (n 为大于 1\ 的正奇数 ),I_{1}=1, $$

$$ I_{n}=\frac{n-1}{n}I_{n-2} $$

$$ 147.I_{n}=\int_{0}^{\frac{\pi}{2}}\sin^{n}x\mathrm{d}x=\int_{0}^{\frac{\pi}{2}}\cos x\mathrm{d}x, $$