（补档）

微分公式

$$d C =0\tag{1}$$
$$dx^{\mu}=\mu x^{\mu -1}dx\tag{2}$$
$$da^{x}=a^{x}\ln a dx\tag{3}$$
$$de^{x}=e^{x}dx\tag{4}$$
$$d\log _{a}x=\frac{1}{x\ln a}dx\tag{5}$$
$$d\ln x=\frac{1}{x}dx\tag{6}$$
$$d\sin x=\cos xdx\tag{7}$$
$$d\cos x=-\sin xdx\tag{8}$$
$$d\tan x=\sec ^{2}xdx\tag{9}$$
$$d\cot x=-\csc ^{2}xdx\tag{10}$$
$$d\sec x=\sec x\tan xdx\tag{11}$$
$$d\csc x=-\csc x\cot xdx\tag{12}$$
$$d\arcsin x=\frac{1}{\sqrt{1-x^{2}}}dx\tag{13}$$
$$d\arccos x=-\frac{1}{\sqrt{1-x^{2}}}dx\tag{14}$$
$$d\arctan x=\frac{1}{1+x^{2}}dx\tag{15}$$
$$d arccot x=-\frac{1}{1+x^{2}}dx\tag{16}$$

积分公式

$$\int kdx=kx+C\tag{1}$$
$$\int x^{\mu}dx=\frac{1}{\mu +1}x^{\mu +1}+C\tag{2}$$
$$\int \frac{1}{x}dx=\ln|x|+C\tag{3}$$
$$\int \frac{1}{1+x^{2}}dx=\arctan x+C\tag{4}$$
$$\int \frac{1}{\sqrt{1-x^{2}}}dx=\arcsin x+C\tag{5}$$
$$\int \cos xdx=\sin x+C\tag{6}$$
$$\int \sin xdx=-\cos x+C\tag{7}$$
$$\int \sec^{2}xdx=\tan x+C\tag{8}$$
$$\int \csc^{2}xdx=-\cot x+C\tag{9}$$
$$\int \sec x\tan xdx=\sec x+C\tag{10}$$
$$\int \csc x \cot xdx=-\csc x+C\tag{11}$$
$$\int e^{x}dx=e^{x}+C\tag{12}$$
$$\int a^{x}dx=\frac{a^{x}}{\ln a}+C\tag{13}$$
$$\int \sinh xdx=\cosh x+C\tag{14}$$
$$\int \cosh xdx=\sinh x+C\tag{15}$$
$$\int \tan xdx=-\ln |\cos x|+C\tag{16}$$
$$\int \cot xdx=\ln |\sin x|+C\tag{17}$$
$$\int \sec xdx=\ln |\sec x+\tan x|+C\tag{18}$$
$$\int \csc xdx=\ln |\csc x- \cot x|+C\tag{19}$$
$$\int \frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan \frac{x}{a}+C\tag{20}$$
$$\int \frac{1}{x^{2}-a^{2}}dx=\frac{1}{2a}\ln | \frac{x-a}{x+a}|+C\tag{21}$$
$$\int \frac{1}{\sqrt{a^{2}-x^{2}}}dx=\arcsin \frac{x}{a}+C\tag{22}$$
$$\int \frac{1}{\sqrt{x^{2}+a^{2}}}dx=\ln (x+\sqrt{x^{2}+a^{2}})+C\tag{23}$$
$$\int \frac{1}{\sqrt{x^{2}-a^{2}}}dx=\ln |x+\sqrt{x^{2}-a^{2}}|+C\tag{24}$$
$$\int \sqrt{a^{2}-x^{2}}dx=\frac{a^{2}}{2}\arcsin \frac{x}{a}+\frac{x}{2}\sqrt{a^{2}-x^{2}}+C\tag{25}$$

无穷小代换

$$\sin x\sim x\tag{1}$$
$$\tan x\sim x\tag{2}$$
$$\arcsin x \sim x\tag{3}$$
$$\arctan x \sim x\tag{4}$$
$$\ln (1+x)\sim x\tag{5}$$
$$e^{x}-1\sim x\tag{6}$$
$$(1+x)^{\alpha}-1\sim \alpha x\tag{7}$$

三角

$$\sin^{2}\alpha+\cos^{2}\alpha = 1$$
$$1+\tan^{2}\alpha=\sec^{2}\alpha$$
$$1+\cot^{2}\alpha=\csc^{2}\alpha$$

Maclaurin公式

$$\sin x =x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\cdot\cdot\cdot+(-1)^{m-1}\frac{x^{2m-1}}{(2m-1)!}+R_{2m}(x)\tag{1}$$
$$\cos x =1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdot\cdot\cdot+(-1)^{m}\frac{x^{2m}}{(2m)!}+R_{2m+1}(x)\tag{1}$$