常见积分表

• 常数 $$\int kdx = kx + C$$
• 多项式

  • $$\int x^\alpha dx = \dfrac{x^{\alpha+1}}{\alpha+1} + C \space (\alpha \neq -1)$$

  • $$\int \dfrac{dx}{x} = \ln |x| + C \space (\alpha = -1)$$

• 指数

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

  • $$\int e^x dx = e^x + C$$

• 三角函数

  • $$\int \sin x dx = -\cos x + C$$

  • $$\int \cos x dx = \sin x + C$$

  • $$\int \dfrac{1}{\cos^2 x} dx = \tan x + C$$

  • $$\int \dfrac{1}{\sin^2 x} = -\cot x + C$$

  • $$\int \dfrac{\sin x}{\cos^2 x} dx = \dfrac{1}{\cos x} + C$$

  • $$\int \dfrac{\cos x}{\sin^2 x} dx = -\dfrac{1}{\sin x} + C$$

  • $$\int \sin^2 x dx = \dfrac{x - \sin x \cos x}{2}$$

  • $$\int \cos^2 x dx = \dfrac{x + \sin x \cos x}{2} + C$$

  • $$\int \dfrac{1}{\sin x} dx = \ln |\dfrac{1}{\sin x} - \dfrac{1}{\tan x}| + C$$

  • $$\int \dfrac{1}{\cos x} dx = \ln |\dfrac{1}{\cos x} - \tan x| + C$$

• 分母平方带常数

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

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

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

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

  • $$\int \dfrac{dx}{x^2 +a ^2} = \dfrac{1}{a} \arcsin \dfrac{1}{a} + C$$