幂指函数

$$\int k dx = kx + C$$ $$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$ $$\int \frac{1}{x} dx = \ln|x| + C$$ $$\int e^x dx = e^x + C$$ $$\int a^x dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1)$$

三角函数

$$\int \sin x dx = -\cos x + C$$ $$\int \cos x dx = \sin x + C$$ $$\int \tan x dx = -\ln|\cos x| + C$$ $$\int \cot x dx = \ln|\sin x| + C$$ $$\int \sec x dx = \ln|\sec x + \tan x| + C$$ $$\int \csc x dx = \ln|\csc x - \cot x| + C$$ $$\int \sec^2 x dx = \tan x + C$$ $$\int \csc^2 x dx = -\cot x + C$$ $$\int \frac{1}{\sin^2 x} dx = -\cot x + C$$ $$\int \frac{1}{1 - \sin x} dx = -\ln|1 - \sin x - \cos x| + C$$

反三角函数

$$\int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C$$ $$\int \frac{1}{\sqrt{a^2 - x^2}} dx = \arcsin \frac{x}{a} + C \quad (a > 0)$$ $$\int -\frac{1}{\sqrt{1-x^2}} dx = \arccos x + C$$ $$\int \frac{1}{1+x^2} dx = \arctan x + C$$ $$\int -\frac{1}{x^2+1} dx = \text{arccot} \thinspace x + C$$ $$\int \frac{1}{x\sqrt{x^2-1}} dx = \text{arcsec} \thinspace x + C$$ $$\int -\frac{1}{x\sqrt{x^2-1}} dx = \text{arccsc} \thinspace x + C$$

反双曲函数

$$ \int \frac{1}{\sqrt{x^2 + a^2}} \thinspace dx = \ln|x + \sqrt{x^2 + a^2}| + C = \text{arcsinh}\left(\frac{x}{a}\right) + C $$

$$ \int \frac{1}{\sqrt{x^2 - a^2}} \thinspace dx = \ln|x + \sqrt{x^2 - a^2}| + C = \text{arccosh}\left(\frac{x}{a}\right) + C $$

$$ \int \frac{1}{1 - x^2} \thinspace dx = \frac{1}{2} \ln\left|\frac{1 + x}{1 - x}\right| + C = \text{arctanh}(x) + C \quad (|x| < 1) $$

$$ \int \frac{1}{x^2 - 1} \thinspace dx = \frac{1}{2} \ln\left|\frac{x - 1}{x + 1}\right| + C = \text{arccoth}(x) + C \quad (|x| > 1) $$

$$ \int \frac{1}{x \sqrt{1 - x^2}} \thinspace dx = -\ln\left|x + \sqrt{x^2 - 1}\right| + C = \text{arcsech}(x) + C \quad (0 < x < 1) $$

$$ \int \frac{1}{|x| \sqrt{1 + x^2}} \thinspace dx = \ln\left|x + \sqrt{x^2 + 1}\right| + C = \text{arccsch}(x) + C $$

反圆锥曲线函数与对数函数的统一 反圆锥曲线函数与对数函数的统一#总结