高数常用公式

求导公式

$$\begin{aligned} &\frac{d}{dx}(c) = 0,\\ &\frac{d}{dx}(x^n) = nx^{n-1},\\ &\frac{d}{dx}(\sin x) = \cos x, &&\frac{d}{dx}(\cos x) = -\sin x\\ &\frac{d}{dx}(\tan x) = \sec^2 x, &&\frac{d}{dx}(\cot x) = -\csc^2 x\\ &\frac{d}{dx}(\sec x) = \sec x \tan x, &&\frac{d}{dx}(\csc x) = -\csc x \cot x\\ &\frac{d}{dx}(\ln x) = \frac{1}{x}, &&\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}, \ a > 0, a \neq 1\\ &\frac{d}{dx}(e^x) = e^x \end{aligned}$$

积分公式

$$\begin{aligned} &\int c dx = cx + C, \\ &\int x^n dx = \frac{x^{n+1}}{n+1} + C,\ n \neq -1\\ &\int \sin x dx = -\cos x + C, &&\int \cos x dx = \sin x + C\\ &\int \tan x dx = -\ln |\cos x| + C, &&\int \cot x dx = \ln |\sin x| + C\\ &\int \sec x dx = \ln |\sec x + \tan x| + C, &&\int \csc x dx = -\ln |\csc x + \cot x| + C\\ &\int \frac{1}{x} dx = \ln |x| + C, &&\int \frac{1}{x^2} dx = -\frac{1}{x} + C\\ &\int e^x dx = e^x + C, &&\int a^x dx = \frac{a^x}{\ln a} + C, \ a > 0, a \neq 1 \end{aligned}$$