基本积分表共24个公式

在微积分中，基本积分表是非常重要的工具，以下为你列举常见的 24 个基本积分公式：

$\int kdx = kx + C$ （$k$ 为常数）

$\int x^{\mu}dx = \frac{x^{\mu + 1}}{\mu + 1} + C (\mu \neq - 1)$

$\int \frac{1}{x}dx = \ln |x| + C$

$\int a^{x}dx = \frac{a^{x}}{\ln a} + C (a > 0,a \neq 1)$

$\int e^{x}dx = e^{x} + C$

$\int \sin xdx = - \cos x + C$

$\int \cos xdx = \sin x + C$

$\int \sec ^{2}xdx = \tan x + C$

$\int \csc ^{2}xdx = - \cot x + C$

$\int \sec x\tan xdx = \sec x + C$

$\int \csc x\cot xdx = - \csc x + C$

$\int \frac{1}{\sqrt{1 - x^{2}}}dx = \arcsin x + C = - \arccos x + C_1$

​1​dx=arcsinx+C=−arccosx+C1​

$\int \frac{1}{1 + x^{2}}dx = \arctan x + C = - \text{arccot}x + C_1$

$\int \tan xdx = - \ln|\cos x| + C$

$\int \cot xdx = \ln|\sin x| + C$

$\int \sec xdx = \ln|\sec x + \tan x| + C$

$\int \csc xdx = \ln|\csc x - \cot x| + C$

$\int \frac{1}{x^{2} - a^{2}}dx = \frac{1}{2a}\ln|\frac{x - a}{x + a}| + C$

$\int \frac{1}{a^{2} - x^{2}}dx = \frac{1}{2a}\ln|\frac{a + x}{a - x}| + C$

$\int \frac{1}{\sqrt{x^{2} + a^{2}}}dx = \ln(x + \sqrt{x^{2} + a^{2}}) + C$

​1​dx=ln(x+x2+a2

​)+C

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

​1​dx=ln∣x+x2−a2

​∣+C

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

​dx=2x​a2−x2

​+2a2​arcsinax​+C

$\int \sqrt{x^{2} + a^{2}}dx = \frac{x}{2}\sqrt{x^{2} + a^{2}} + \frac{a^{2}}{2}\ln(x + \sqrt{x^{2} + a^{2}}) + C$

​dx=2x​x2+a2

​+2a2​ln(x+x2+a2

​)+C

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

​dx=2x​x2−a2

​−2a2​ln∣x+x2−a2

​∣+C

这些公式是积分运算的基础，在求解各种积分问题时经常会用到。