目录！

目录！等价无穷小$(x\to0)$三角函数函数关系诱导公式导数公式泰勒公式泰勒定理常用的泰勒展开式基本积分积分换元法常用凑积分形式积分表（一）含有$ax+b$的积分（二）含有$\sqrt{ax+b}$的积分（三）含有$x^2\pm a^2$的积分（四）含有$ax^2+b\ \ (a>0)$的积分（五）含有$ax^2+bx+c\ (a>0)$的积分（六）含有$\sqrt{x^2+a^2}\ (a>0)$的积分（七）含有$\sqrt{x^2-a^2}\ (a>0)$的积分（八）含有$\sqrt{a^2-x^2}\ (a>0)$的积分（九）含有$\sqrt{\pm ax^2+bx+c}\ (a>0)$的积分（十）含有$\sqrt{\pm \frac{x-a}{x-b}}$或$\sqrt{(x-a)(b-a)}$的积分（十一）含有三角函数的积分（十二）含有反三角函数的积分（其中a>0）（十三）含有指数函数的积分（十四）含有对数函数的积分（十五）含有双曲函数的积分（十六）定积分牛顿-莱布尼兹公式

等价无穷小$(x\to 0)$$(x\to0)$

$\sin x\sim x\arcsin x\sim x$$\sin x \sim x\ \ \ \ \ \ \arcsin x\sim x$

$\tan x\sim x\arctan x\sim x$$\tan x\sim x\ \ \ \ \ \ \arctan x\sim x$

$e^x-1\sim x{a}^x-1\sim x\ln a$$e^x-1\sim x\ \ \ \ \ \ a^x-1\sim x\ln a$

$\ln (1+x)\sim x{\log }_a(1+x)\sim x/\ln a$$\ln(1+x)\sim x\ \ \ \ \ \ \log_a(1+x)\sim x/\ln a$

$1-\cos x\sim 1/2^2$$1-\cos x\sim 1/2 ^2$

$x-\ln (1+x)\sim 1/2x^2$$x-\ln(1+x)\sim1/2\ x^2$

$(1+x{)}^a-1\sim ax$$(1+x)^a-1\sim ax$

$(1+bx{)}^a-1\sim abx$$(1+bx)^a-1\sim abx$

$x-\sin x\sim \arcsin x-x\sim 1/6x^3$$x-\sin x\sim \arcsin x-x\sim1/6\ x^3$

$\tan x-x\sim x-\arctan x\sim 1/3x^3$$\tan x-x\sim x-\arctan x\sim 1/3\ x^3$

$\tan x-\sin x\sim \arcsin x-\arctan x\sim 1/2x^3$$\tan x-\sin x\sim \arcsin x-\arctan x\sim 1/2 \ x^3$

三角函数

函数关系

• 倒数关系

  $\sin \alpha \cdot \csc \alpha =1$$\sin{\alpha}\cdot\csc{\alpha} = 1$

  $\cos \alpha \cdot \sec \alpha =1$$\cos{\alpha}\cdot\sec{\alpha} = 1$

  $\tan \alpha \cdot \cot \alpha =1$$\tan{\alpha}\cdot \cot{\alpha} = 1$

• 商数关系

  $\tan \alpha =\frac{\sin \alpha }{\cos \alpha }$$\tan{\alpha}=\frac{\sin{\alpha}}{\cos{\alpha}}$

  $\cot \alpha =\frac{\cos \alpha }{\sin \alpha }$$\cot{\alpha} = \frac{\cos{\alpha}}{\sin{\alpha}}$

• 平方关系

  ${\sin }^2\alpha +{\cos }^2\alpha =1$$\sin^2\alpha+\cos^2\alpha=1$

  $1+{\tan }^2\alpha ={\sec }^2\alpha$$1+\tan^2\alpha=\sec^2\alpha$

  $1+{\cot }^2\alpha ={\csc }^2\alpha$$1+\cot^2 \alpha= \csc^2\alpha$

诱导公式

• $f(\frac{k}{2}\pi )=g(\alpha ),k\in Z$$f(\frac{k}{2}\pi)=g(\alpha),k\in Z$

  奇变偶不变，符号看象限

导数公式

• $(c{)}^{\prime }=0$$(c)' = 0$

• $(x^{\mu }{)}^{\prime }=\mu x^{\mu -1}$$(x^\mu)'=\mu x^{\mu-1}$

• $(\sin x{)}^{\prime }=\cos x$$(\sin x)'=\cos x$

• $(\cos x{)}^{\prime }=-\sin x$$(\cos x)'=-\sin x$

• $(\tan x{)}^{\prime }={\sec }^{2}x$$(\tan x)'=\sec^2x$

• $(\cot x{)}^{\prime }=-{\csc }^{2}x$$(\cot x)'=-\csc^2x$

• $(\sec x{)}^{\prime }=\sec x\cdot \tan x$$(\sec x)'=\sec x\cdot \tan x$

• $(\csc x{)}^{\prime }=-\csc x\cdot \cot x$$(\csc x)'=-\csc x\cdot \cot x$

• $(a^x{)}^{\prime }=a^x\ln a$$(a^x)'=a^x\ln a$

• $(e^x{)}^{\prime }=e^x$$(e^x)'=e^x$

• $({\log }_{a}x{)}^{\prime }=\frac{1}{x\ln a}$$(\log_ax)'=\frac{1}{x\ln a}$

• $(\ln x{)}^{\prime }=\frac{1}{x}$$(\ln x)'=\frac{1}{x}$

• $(\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}}$$(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$

• $(\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}$$(\arccos x)'=-\frac{1}{\sqrt{1-x^2}}$

• $(\arctan x{)}^{\prime }=\frac{1}{1+x^2}$$(\arctan x)'=\frac{1}{1+x^2}$

• $(\mathrm{arccot}x{)}^{\prime }=-\frac{1}{1+x^2}$$(\arccot x)'=-\frac{1}{1+x^2}$

泰勒公式

泰勒定理

$f(x)={\displaystyle \sum _{k=0}^n\frac{f^k(x_0)}{k!}(x-x_0)+\frac{f^{n+1}(\xi )}{(n+1)!}(x-x_0{)}^{n+1}}$$f(x)= \displaystyle\sum^{n}_{k =0}\frac{f^k(x_0)}{k!}(x-x_0)+\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_0)^{n+1}$

$f(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{f^{″}(x_0)}{2!}(x-x_0{)}^2+\cdots +\frac{f^{(n)}(x_0)}{n!}+R_n(x)$$f(x) = f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\dots+\frac{f^{(n)}(x_0)}{n!}+R_n(x)$

• 佩亚诺型余项： $R_n(x)=o((x-x_0{)}^n)$$R_n(x) = o((x-x_0)^n)$
• 拉格朗日型余项： $R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-x_0{)}^{n+1}$$R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}$

• 麦克劳林公式

  $f(x)={\displaystyle \sum _{k=0}^n\frac{f^{(k)}(0)}{k!}{x}^k+\frac{f^{n+1}(\xi )}{(n+1)!}}$$f(x)= \displaystyle\sum^{n}_{k =0} \frac{f^{(k)}(0)}{k!}x^k+\frac{f^(\xi)}{(n+1)!}$

常用的泰勒展开式

基本积分

积分换元法

• 第一类换元法

  $$\int f[\phi (x)]\cdot {\phi }^{\prime }(x)=[\int f(u)du{]}_{u=\phi (x)}$$

• 第二类换元法

  $$\int f(x)dx=[\int f[\psi (t)]{\psi }^{\prime }(t)dt{]}_{t={\psi }^{-1}(x)}$$

   

常用凑积分形式

• $$dx=\frac{1}{a}d(ax+b)$$
• $$x^{m}dx=\frac{d(x^{m+1})}{m+1}=\frac{d(a{x}^{m+1}+b)}{(m+1)a}$$
• $$\frac{1}{x^2}dx=-d\frac{1}{x}$$
• $$\frac{dx}{\sqrt{x}}=2d\sqrt{x}$$
• $$\frac{dx}{\sqrt{1-x^2}}=d\arcsin x$$
• $$\frac{dx}{1-x^2}=d\arctan x$$
• $$\frac{dx}{{\cos }^{2}x}=d\tan x$$
• $$e^{x}dx=d{e}^x$$
• $$\frac{dx}{x}=d\ln x$$
• $$\int \frac{f^{\prime }(x)}{f(x)}dx=\int \frac{df(x)}{f(x)}=\ln f(x)+C$$

积分表

（一）含有$ax+b$$ax+b$的积分

• $$\int \frac{dx}{ax+b}=\frac{1}{a}\ln |ax+b|+C$$
• $$\int (ax+b{)}^{\mu }dx=\frac{1}{a(\mu +1)}(ax+b{)}^{\mu +1}+C(\mu \ne 1)$$
• $$\int \frac{x}{ax+b}dx=\frac{1}{a^2}(ax+b-b\ln |ax+b|)+C$$
• $$\int \frac{x^2}{ax+b}dx=\frac{1}{a^3}[\frac{1}{2}(ax+b{)}^2-2b(ax+b)+b^2\ln |ax+b|]+C$$
• $$\int \frac{dx}{x(ax+b)}=-\frac{1}{b}\ln |\frac{ax+b}{x}|+C$$
• $$\int \frac{dx}{x^2(ax+b)}=-\frac{1}{bx}+\frac{a}{b^2}\ln |\frac{ax+b}{x}|+C$$
• $$\int \frac{x}{(ax+b{)}^2}dx=\frac{1}{a^2}(\ln |ax+b|+\frac{b}{ax+b})+C$$
• $$\int \frac{x^2}{(ax+b{)}^2}dx=\frac{1}{a^3}(ax+b-b\ln |ax+b|-\frac{b^2}{ax+b})$$
• $$\int \frac{dx}{x(ax+b{)}^2}=\frac{1}{b(ax+b)}-\frac{1}{b^2}\ln |\frac{ax+b}{x}|+C$$

（二）含有$\sqrt{ax+b}$$\sqrt{ax+b}$的积分

• $$\int \sqrt{ax+b}dx=\frac{2}{3a}\sqrt{(ax+b{)}^3}+C$$
• $$\int x\sqrt{ax+b}dx=\frac{2}{15a^2}(3ax-2b)\sqrt{(ax+b{)}^3}+C$$
• $$\int x^2\sqrt{ax+b}dx=\frac{2}{105a^3}(15a^2{x}^2-12abx+8b^2)\sqrt{(ax+b{)}^3}+C$$
• $$\int \frac{x}{\sqrt{ax+b}}dx=\frac{2}{3a^2}(ax-2b)\sqrt{ax+b}+C$$
• $$\int \frac{x^2}{\sqrt{ax+b}}dx=\frac{2}{15a^3}(3a^2{x}^2-4abx+8b^2)\sqrt{ax+b}+C$$
• $$\int \frac{dx}{x\sqrt{ax+b}}=\{\begin{array}{l}\frac{1}{\sqrt{b}}\ln |\frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}}|+C(b>0),\\ \frac{2}{-\sqrt{-b}}\arctan \sqrt{\frac{ax+b}{-b}}+C(b<0).\end{array}$$
• $$\int \frac{dx}{x^2\sqrt{ax+b}}=-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}\int \frac{dx}{x\sqrt{ax+b}}$$
• $$\int \frac{\sqrt{ax+b}}{x}dx=2\sqrt{ax+b}+b\int \frac{dx}{x\sqrt{ax+b}}$$
• $$\int \frac{\sqrt{ax+b}}{x^2}dx=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int \frac{dx}{x\sqrt{ax+b}}$$

（三）含有$x^2\pm a^2$$x^2\pm a^2$的积分

• $$\int \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan \frac{x}{a}+C$$
• $$\int \frac{dx}{(x^2+a^2{)}^n}=\frac{x}{2(n-1)a^2(x^2+a^2{)}^{n-1}}+\frac{2n-3}{2(n-1)a^2}\int \frac{dx}{(x^2+a^2{)}^{n-1}}$$
• $$\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$$

（四）含有$a{x}^2+b(a>0)$$ax^2+b\ \ (a>0)$的积分

• $$\int \frac{dx}{a{x}^2+b}=\{\begin{array}{l}\frac{1}{\sqrt{ab}}\arctan \sqrt{\frac{a}{b}}x+C(b>0),\\ \frac{1}{2\sqrt{-ab}}\ln |\frac{\sqrt{a}x-\sqrt{-b}}{\sqrt{a}x+\sqrt{-b}}|+C(b<0).\end{array}$$
• $$\int \frac{x}{a{x}^2+b}dx=\frac{1}{2a}\ln |a{x}^2+b|+C$$
• $$\int \frac{x^2}{a{x}^2+b}dx=\frac{x}{a}-\frac{b}{a}\int \frac{dx}{a{x}^2+b}$$
• $$\int \frac{dx}{x(a{x}^2+b)}=\frac{1}{2b}\ln \frac{x^2}{|a{x}^2+b|}+C$$
• $$\int \frac{dx}{x^2(a{x}^2+b)}=-\frac{1}{bx}-\frac{a}{b}\int \frac{dx}{a{x}^2+b}$$
• $$\int \frac{dx}{x^3(a{x}^2+b)}=\frac{a}{2b^2}\ln \frac{|a{x}^2+b|}{x^2}-\frac{1}{2b{x}^2}+C$$
• $$\int \frac{dx}{(a{x}^2+b{)}^2}=\frac{x}{2b(a{x}^3+b)}+\frac{1}{2b}\int \frac{dx}{a{x}^2+b}$$

（五）含有$a{x}^2+bx+c(a>0)$$ax^2+bx+c\ (a>0)$的积分

• $$\int \frac{dx}{a{x}^2+bx+c}\{\begin{array}{l}\frac{2}{\sqrt{4ac-b^2}}\arctan \frac{2ax+b}{\sqrt{4ac-b^2}}+C(b^2<4ac)\\ \frac{1}{\sqrt{b^2-4ac}}\ln |\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}|+C(b^2>4ac)\end{array}$$
• $$\int \frac{x}{a{x}^2+bx+c}dx=\frac{1}{2a}\ln |a{x}^2+bx+c|-\frac{b}{2a}\int \frac{dx}{a{x}^2+bx+c}$$

   

（六）含有$\sqrt{x^2+a^2}(a>0)$$\sqrt{x^2+a^2}\ (a>0)$的积分

• $$\int \frac{dx}{\sqrt{x^2+a^2}}=\mathrm{arsh}\frac{x}{a}+C_1=\ln (x+\sqrt{x^2+a^2})+C$$
• $$\int \frac{dx}{\sqrt{(x^2+a^2{)}^3}}=\frac{x}{a^2\sqrt{x^2+a^2}}+C$$
• $$\int \frac{x}{\sqrt{x^2+a^2}}dx=\sqrt{x^2+a^2}+C$$
• $$\int \frac{x}{\sqrt{(x^2+a^2{)}^3}}dx=-\frac{1}{\sqrt{x^2+a^2}}+C$$
• $$\int \frac{x^2}{\sqrt{x^2+a^2}}dx=\frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln (x+\sqrt{x^2+a^2})+C$$
• $$\int \frac{x^2}{\sqrt{(x^2+a^2{)}^3}}dx=-\frac{x}{\sqrt{x^2+a^2}}+\ln (x+\sqrt{x^2+a^2})+C$$
• $$\int \frac{dx}{x\sqrt{x^2+a^2}}=\frac{1}{a}\ln \frac{\sqrt{x^2+a^2}-a}{|x|}+C$$
• $$\int \frac{dx}{x^2\sqrt{x^2+a^2}}dx=-\frac{\sqrt{x^2+a^2}}{a^{2}x}+C$$
• $$\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln (x+\sqrt{x^2+a^2})+C$$
• $$\int \sqrt{(x^2+a^2{)}^3}dx=\frac{x}{8}(2x^2+5a^2)\sqrt{x^2+a^2}+\frac{3}{8}{a}^4\ln (x+\sqrt{x^2+a^2})+C$$
• $$\int x\sqrt{x^2+a^2}dx=\frac{1}{3}\sqrt{(x^2+a^2{)}^3}+C$$
• $$\int x^2\sqrt{x^2+a^2}dx=\frac{x}{8}(2x^2+a^2)\sqrt{x^2+a^2}-\frac{a^4}{8}\ln (x+\sqrt{x^2+a^2})+C$$
• $$\int \frac{\sqrt{x^2+a^2}}{x}dx=\sqrt{x^2+a^2}+a\ln \frac{\sqrt{x^2+a^2}-a}{|x|}+C$$
• $$\int \frac{\sqrt{x^2+a^2}}{x^2}dx=-\frac{\sqrt{x^2+a^2}}{x}+\ln (x+\sqrt{x^2+a^2})+C$$

（七）含有$\sqrt{x^2-a^2}(a>0)$$\sqrt{x^2-a^2}\ (a>0)$的积分

• $$\int \frac{dx}{\sqrt{x^2-a^2}}=\frac{x}{|x|}\mathrm{arsh}\frac{|x|}{a}+C_1+\ln |x+\sqrt{x^2-a^2}|+C$$
• $$\int \frac{dx}{\sqrt{(x^2-a^2{)}^3}}=-\frac{x}{\sqrt{x^2-a^2}}+C$$
• $$\int \frac{x}{\sqrt{x^2-a^2}}dx=\sqrt{x^2-a^2}+C$$
• $$\int \frac{x}{\sqrt{(x^2-a^2{)}^3}}dx=-\frac{1}{\sqrt{x^2-a^2}}+C$$
• $$\int \frac{x^2}{\sqrt{x^2-a^2}}dx=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\ln |x+\sqrt{x^2-a^2}|+C$$
• $$\int \frac{x^2}{\sqrt{(x^2-a^2{)}^3}}dx=-\frac{x}{\sqrt{x^2-a^2}}+\ln |x+\sqrt{x^2-a^2}|+C$$
• $$\int \frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}\arccos \frac{a}{|x|}+C$$
• $$\int \frac{dx}{x^2\sqrt{x^2-a^2}}=\frac{\sqrt{x^2-a^2}}{a^{2}x}+C$$
• $$\int \sqrt{x\dots \dots 2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln |x+\sqrt{x^2-a^2}|+C$$
• $$\int \sqrt{(x^2-a^2{)}^3}dx=\frac{x}{8}(2x^2-5a^2)\sqrt{x^2-a^2}+\frac{3}{8}{a}^4\ln |x+\sqrt{x^2-a^2}|+C$$
• $$\int x\sqrt{x^2-a^2}dx=\frac{1}{3}\sqrt{(x^2-a^2{)}^3}+C$$
• $$\int x^2\sqrt{x^2-a^2}dx=\frac{x}{8}(2x^2-a^2)\sqrt{x^2-a^2}-\frac{a^4}{8}\ln |x+\sqrt{x^2-a^2}|+C$$
• $$\int \frac{\sqrt{x^2-a^2}}{x}dx=\sqrt{x^2-a^2}-a\arccos \frac{a}{|x|}+C$$
• $$\int \frac{\sqrt{x^2-a^2}}{x^2}dx=-\frac{\sqrt{x^2-a^2}}{x}+\ln |x+\sqrt{x^2-a^2}|+C$$

（八）含有$\sqrt{a^2-x^2}(a>0)$$\sqrt{a^2-x^2}\ (a>0)$的积分

挖坑待填

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

挖坑待填

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

• $$\int \sqrt{\frac{x-a}{x-b}}dx=(x-b)\sqrt{\frac{x-a}{x-b}}=(b-a)\ln (\sqrt{|x-a|}+\sqrt{|x-b|})+C$$
• $$\int \sqrt{\frac{x-a}{b-x}}dx=(x-b)\sqrt{\frac{x-a}{b-x}}+(b-a)\arcsin \sqrt{\frac{x-a}{b-a}}+C$$
• $$\int \frac{dx}{\sqrt{(x-a)(b-x)}}=2\arcsin \sqrt{\frac{x-a}{b-a}}+C(a<b)$$
• $$\int \sqrt{(x-a)(b-x)}dx=\frac{2x-a-b}{4}\sqrt{(x-a)(b-x)}+\frac{(b-a{)}^2}{4}\arcsin \sqrt{\frac{x-a}{b-a}}+C(a<b)$$

（十一）含有三角函数的积分

• $$\int \sin xdx=-\cos x+C$$
• $$\int \cos xdx=\sin x+C$$
• $$\int \tan xdx=-\ln |\cos x|+C$$
• $$\int \cot xdx=\ln |\sin x|+C$$
• $$\int \sec xdx=\ln |\tan \left(\frac{\pi }{4}\right)+\frac{x}{2}|+C=\ln |\sec x+\tan x|+C$$
• $$\int \csc xdx=\ln |\tan \frac{x}{2}|+C+\ln |\csc x-\cot x|+C$$
• $$\int {\sec }^{2}xdx=\tan x+C$$
• $$\int \sec \left(x\right)=\ln (\sec \left(x\right)+\tan \left(x\right))$$

•  

• $$\int {\csc }^{2}xdx=-\cot x+C$$
• $$\int \sec x\tan xdx=\sec x+C$$
• $$\int \csc x\cot xdx=-\csc x+C$$
• $$\int {\sin }^{2}xdx=\frac{x}{2}-\frac{1}{4}\sin 2x+C$$
• $$\int {\cos }^{2}xdx=\frac{x}{2}+\frac{1}{4}\sin 2x+C$$
• $$\int {\sin }^{n}xdx=-\frac{1}{n}{\sin }^{n-1}x\cos x+\frac{n-1}{n}\int {\sin }^{n-2}xdx$$
• $$\int {\cos }^{n}xdx=\frac{1}{n}{\cos }^{n-1}x\sin x+\frac{n-1}{n}\int {\cos }^{n-2}xdx$$
• $$\int \frac{dx}{{\sin }^{n}x}=-\frac{1}{n-1}\cdot \frac{\cos x}{{\sin }^{n-1}x}+\frac{n-2}{n-1}\int \frac{dx}{{\sin }^{n-2}x}$$
• $$\int \frac{dx}{{\cos }^{n}x}=\frac{1}{n-1}\cdot \frac{\sin x}{{\cos }^{n-1}x}+\frac{n-2}{n-1}\int \frac{dx}{{\cos }^{n-2}x}$$
• $$\begin{array}{c}\int {\cos }^{m}x{\sin }^{n}xdx=\frac{1}{m+n}{\cos }^{m-1}x{\sin }^{n+1}x+\frac{m-1}{m+n}\int {\cos }^{m-2}x{\sin }^{n}xdx\\ =-\frac{1}{m+n}{\cos }^{m+1}x{\sin }^{n-1}x+\frac{n-1}{m+n}\int {\cos }^{m}x{\sin }^{n-2}xdx\end{array}$$
• $$\int \sin ax\cos bxdx=-\frac{1}{2(a+b)}\cos (a+b)x-\frac{1}{2(a-b)\cos (a-b)x+C}$$
• $$\int \sin ax\sin bxdx=-\frac{1}{2(a+b)}\sin (a+b)x+\frac{1}{2(a-b)}\sin (a-b)x+C$$
• $$\int cosax\cos bxdx=\frac{1}{2(a+b)}\sin (a+b)x+\frac{1}{2(a-b)}\sin (a-b)x+C$$
• $$\int \frac{dx}{a+b\sin x}=\frac{2}{\sqrt{a^2-b^2}}\arctan \frac{a\tan \frac{x}{2}+b}{\sqrt{a^2-b^2}}+C(a^2>b^2)$$
• $$\int \frac{dx}{a+b\sin x}=\frac{1}{\sqrt{a^2-b^2}}\ln |\frac{a\tan \frac{x}{2}+b-\sqrt{b^2-a^2}}{a\tan \frac{x}{2}+b+\sqrt{b^2-a^2}}|+C(a^2<b^2)$$
• $$\int \frac{dx}{a+b\cos x}=\frac{2}{a+b}\sqrt{\frac{a+b}{a-b}}\arctan (\sqrt{\frac{a-b}{a+b}}\tan \frac{x}{2})+C(a^2>b^2)$$
• $$\int \frac{dx}{a+b\cos x}=\frac{1}{a+b}\sqrt{\frac{a+b}{a-b}}\ln |\frac{\tan \frac{x}{2}+\sqrt{\frac{a+b}{a-b}}}{\tan \frac{x}{2}-\sqrt{\frac{a+b}{a-b}}}|+C(a^2<b^2)$$
• $$\int \frac{dx}{a^{2}co{s}^{2}x+b^2{\sin }^{2}x}=\frac{1}{ab}\arctan (\frac{b}{a}\tan x)+C$$
• $$\int \frac{dx}{a^2{\cos }^2-b^2{\sin }^{2}x}=\frac{1}{2ab}\ln |\frac{b\tan x+a}{b\tan x-a}|+C$$
• $$\int x\sin axdx==-\frac{1}{a^2}\sin ax-\frac{1}{a}x\cos ax+C$$
• $$\int x^2\sin axdx=-\frac{1}{a}{x}^2\cos ax+\frac{2}{a^2}x\sin ax+\frac{2}{a^3}\cos ax+C$$
• $$\int x\cos axdx=\frac{1}{a^2}\cos ax+\frac{1}{a}x\sin ax+C$$
• $$\int x^2\cos axdx=\frac{1}{a}{x}^2\sin ax+\frac{2}{a^2}x\cos ax-\frac{2}{a^2}\sin ax+C$$

（十二）含有反三角函数的积分（其中a>0）

• $$\int \arcsin \frac{x}{a}dx=x\arcsin \frac{x}{a}+\sqrt{a^2-x^2}+C$$
• $$\int x\arcsin \frac{x}{a}dx=(\frac{x^2}{2}-\frac{a^2}{4})\arcsin \frac{x}{a}+\frac{x}{4}\sqrt{a^2-x^2}+C$$
• $$\int x^2\arcsin \frac{x}{a}dx=\frac{x^3}{3}\arcsin \frac{x}{a}+\frac{1}{9}(x^2+2a^2)\sqrt{a^2-x^2}+C$$
• $$\int \arccos \frac{x}{a}dx=x\arccos \frac{x}{a}-\sqrt{a^2-x^2}+C$$
• $$\int x\arccos \frac{x}{a}dx=(\frac{x^2}{2}-\frac{a^2}{4})\arccos \frac{x}{a}-\frac{x}{4}\sqrt{a^2-x^2}+C$$
• $$\int x^2\arccos \frac{x}{a}dx=\frac{x^3}{3}\arccos \frac{x}{a}-\frac{1}{9}(x^2+2a^2)\sqrt{a^2-x^2}+C$$
• $$\int \arctan \frac{x}{a}dx=x\arctan \frac{x}{a}-\frac{a}{2}\ln (a^2+x^2)+C$$
• $$\int x\arctan \frac{x}{a}dx=\frac{1}{2}(a^2+x^2)\arctan \frac{x}{a}-\frac{a}{2}x+C$$
• $$\int x^2\arctan \frac{x}{a}dx=\frac{x^3}{3}\arctan \frac{x}{a}-\frac{a}{6}{x}^2+\frac{a^3}{6}\ln (a^2+x^2)+C$$

（十三）含有指数函数的积分

（十四）含有对数函数的积分

（十五）含有双曲函数的积分

（十六）定积分

• $${\int }_{-\pi }^{\pi }\cos nxdx={\int }_{-\pi }^{\pi }\sin nxdx=0$$
• $${\int }_{-\pi }^{\pi }\cos mx\sin nxdx=0$$
• $$\begin{array}{c}{\int }_{-\pi }^{\pi }\cos mx\cos nxdx=\{\begin{array}{l}0,m\ne n,\\ \pi ,m=n.\end{array}\end{array}$$
• $$\begin{array}{c}{\int }_{-\pi }^{\pi }\sin mx\sin nxdx=\{\begin{array}{l}0,m\ne n,\\ \pi ,m=n.\end{array}\end{array}$$
• $$\begin{array}{c}{\int }_0^{\pi }\sin mx\sin nxdx={\int }_0^{\pi }\cos mx\cos nxdx=\{\begin{array}{l}0,m\ne n,\\ \frac{\pi }{2},m=n.\end{array}\end{array}$$
• $$\begin{array}{c}{I}_n={\int }_0^{\frac{\pi }{2}}{\sin }^{n}xdx={\int }_0^{\frac{\pi }{2}}{\cos }^{n}xdx,\\ I_n=\frac{n-1}{n}{I}_{n-2}=\{\begin{array}{l}\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \cdots \cdot \frac{4}{5}\cdot \frac{2}{3}(n\text{为}\text{大}\text{于}1\text{的}\text{奇}\text{数}),I_1=1,\\ \frac{n-1}{x}\cdot \frac{n-3}{n-2}\cdot \cdots \cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{\pi }{2}(n\text{为}\text{正}\text{偶}\text{数}),I_0=\frac{\pi }{2}.\end{array}\end{array}$$