【数学基础】这些微积分公式，你记得吗？

微积分必备公式

1. 基础不定积分公式

• $\int kdx=kx+C$
• $\int x^{\mu }dx=\frac{x^{\mu +1}}{\mu +1}+C,\mu \ne -1$
• $\int \frac{dx}{x}=\ln |x|+C$
• $\int \frac{1}{1+x^2}dx=\arctan x+C$
• $\int \frac{1}{\sqrt[]{1-x^2}}dx=\arcsin x+C$
• $\int a^{x}dx=\frac{a^x}{\ln a}+C$
• $\int e^{x}dx=e^x+C$
• $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan \frac{x}{a}+C$
• $\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$
• $\int \cos xdx=\sin x+C$
• $\int \sin xdx=-\cos x+C$
• $\int {\sec }^{2}xdx=\tan x+C$
• $\int {\csc }^{2}xdx=-\cot x+C$
• $\int \tan xdx=-\ln |\cos x|+C$
• $\int \cot xdx=\ln |\sin x|+C$
• $\int \sec xdx=\ln |\sec x+\tan x|+C$
• $\int \csc xdx=\ln |\csc x-\cot x|+C$
• $\int \frac{1}{\sqrt{x^2\pm a^2}}dx=\ln |x+\sqrt{x^2\pm a^2}|+C$
• $\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \frac{x}{a}+C$

2. 基础导数公式

1. $\frac{d}{dx}x=1$
2. $\frac{d}{dx}\sin x=\cos x$
3. $\frac{d}{dx}\cos x=-\sin x$
4. $\frac{d}{dx}\tan x={\sec }^{2}x$
5. $\frac{d}{dx}\sec x=\sec x\tan x$
6. $\frac{d}{dx}\csc x=-\csc x\cot x$
7. $\frac{d}{dx}\cot x=-{\csc }^{2}x$
8. $\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}$
9. $\frac{d}{dx}\arccos x=-\frac{1}{\sqrt{1-x^2}}$
10. $\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$
11. $\frac{d}{dx}{a}^x=a^x\ln a$
12. $\frac{d}{dx}{e}^x=e^x$
13. $\frac{d}{dx}\ln (x)=\frac{1}{x}(x>0)$
14. $\frac{d}{dx}\ln |x|=\frac{1}{x}(x\ne 0)$
15. $\frac{d}{dx}{\log }_a(x)=\frac{1}{x\ln a}$

3. 基本运算法则的应用

1. $\frac{d}{dx}[f(x){]}^2=n[f(x){]}^{n-1}{f}^{\prime }(x)$
2. $\frac{d}{dx}{e}^{f(x)}=f^{\prime }(x)e^{f(x)}$
3. $\frac{d}{dx}\ln [f(x)]=\frac{f^{\prime }(x)}{f(x)}$
4. $\frac{d}{dx}\sin [f(x)]=f^{\prime }(x)\cos [f(x)]$
5. $\frac{d}{dx}\cos [f(x)]=-f^{\prime }(x)\sin [f(x)]$
6. $\frac{d}{dx}\tan [f(x)]=f^{\prime }(x){\sec }^2[f(x)]$
7. $\frac{d}{dx}\sec [f(x)]=f^{\prime }(x)\sec [f(x)]\tan [f(x)]$
8. $\frac{d}{dx}\frac{1}{\tan f(x)}=\frac{f^{\prime }(x)}{1+[f(x){]}^2}$
9. $\frac{d}{dx}f(x{)}^{g(x)}=f(x{)}^{g(x)}\frac{g(x)f^{\prime }(x)}{f(x)}+\ln f(x)g^{\prime }(x)$

4. 极限的性质

• ${\lim }_{x\to a}[cf(x)]=c{\lim }_{x\to a}f(x)$
• ${\lim }_{x\to a}[f(x)\pm g(x)]={\lim }_{x\to a}f(x)\pm {\lim }_{x\to a}g(x)$
• ${\lim }_{x\to a}[f(x)g(x)]={\lim }_{x\to a}f(x){\lim }_{x\to a}g(x)$
• ${\lim }_{x\to a}\frac{f(x)}{g(x)}=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}({\lim }_{x\to a}g(x)\ne 0)$
• ${\lim }_{x\to a}[f(x){]}^n=[{\lim }_{x\to a}f(x){]}^n$