1. 不定积分的概念与性质
- 在区间I上，函数$f(x)$的带有任意常数项的原函数称为$f(x)$（或$f(x)dx$）在区间I上的不定积分，记作$\int f(x)dx$，其中记号$\int$称为积分号，$f(x)$称为被积函数，$f(x)dx$称为被积表达式，x称为积分变量
- 基本积分表
- $\int kdx = kx + C$
- $\int x^{\mu}dx = \frac{x^{\mu + 1}}{\mu + 1} + C$
- $\int\frac{dx}{x} = \ln|x| + C$
- $\int\frac{dx}{1 + x^2} = \arctan x + C$
- $\int\frac{dx}{\sqrt{1 - x^2}} = \arcsin x + C$
- $\int \cos x dx = \sin x + C$
- $\int \sin x dx = -\cos x + C$
- $\int\frac{dx}{\cos^2 x} = \int \sec^2 x dx = \tan x + C$
- $\int \frac{dx}{\sin^2 x} = \int\csc^2 x dx = -\cot x + C$
- $\int \sec x\tan x dx = \sec x + C$
- $\int \csc x \cot x dx = -\csc x + C$
- $\int e^x dx = e^x + C$
- $\int a^x dx = \frac{a^x}{\ln a} + C$
- $\int sh xdx = ch x + C$
- $\int ch x dx = sh x + C$
- $\int \tan x dx = -\ln|\cos x| + C$
- $\int \cot x dx = \ln|\sin x| + C$
- $\int \sec x dx = \ln|\sec x + \tan x| + C$
- $\int \csc x dx = \ln|\csc x - \cot x| + C$
- $\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\frac{x}{a} + C$
- $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln|\frac{x - a}{x + a}| + C$
- $\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\frac{x}{a} + C$
- $\int \frac{dx}{x^2 + a^2} = \ln(x + \sqrt{x^2 + a^2}) + C$
- $\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln|x + \sqrt{x^2 - a^2}| + C$
- 第一类换元法
- 设$f(u)$具有原函数，$u = \phi(x)$可导，则有换元公式$\int f[\phi(x)]\phi’(x)dx = [\int f(u)du_{u = \phi(x)}]$
- 第二类换元法
- 设$x = \psi(t)$是单调的可导函数，并且$\psi’(t) \neq 0$，又设$f[\psi(t)]\psi’(t)$具有原函数，则有换元公式$\int f(x)dx = [\int f[\psi(t)]\psi’(t)dt]_{t = \psi^{-1}(x)}$，其中$\psi^{-1}(x)$是$x = \psi(t)$的反函数
- 分部积分法
- 设函数$u = u(x)$及$v = v(x)$具有连续导数，则两个函数乘积的导数公式为$(uv)’ = u’v + uv’$，移项，得$uv’ = (uv)’ - u’v$。对这个等式两边求不定积分，得，$\int uv’dx = uv - \int u’v dx$