常见积分表

警告

积分一定要记得带常数 $+C$

• 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

  • $$\int \sin x dx = -\cos x + C$$

  • $$\int \cos x dx = \sin x + C$$

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

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

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

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

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

  • $$\int \dfrac{1}{\cos x} dx = \ln |\dfrac{1}{\cos x} + \tan x| + C = \dfrac{1}{2} \ln \left|\dfrac{1+\sin x}{1-\sin x}\right| + C$$

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

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

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

提示

注意 $\tan^2 x$ 与 $\cos^2 x$ 互转 $\dfrac{1}{\cos^2 x} = 1 + \tan^2 x$$\dfrac{1}{\cos^2 x} = (\tan^2 x)'$

• 分母平方带常数

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

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

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

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

提示

根号下 $x^2$ 和 $a^2$ 可使用换元法 由于 $\tan^2 x + 1 = \dfrac{1}{\cos^2 x}$, $\dfrac{1}{\cos^2 x} - 1 = \tan^2 x$$\sqrt{a^2-x^2} \implies x = a\sin t$$\sqrt{a^2+x^2} \implies x = a\tan t$$\sqrt{x^2-a^2} \implies x = \dfrac{a}{\cos t}$

• 根式平方带常数

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

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

• 多项式比根式: 分母用配方消去一次项, 分子可拆成二次与一次的线性组合
  • 分子二次

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

  • 分子一次

    $$\int \dfrac{x dx}{\sqrt{a^2 \pm x^2}} = \pm \dfrac{1}{2} \int \dfrac{d(a^2 \pm x^2)}{\sqrt{a^2 \pm x^2}} = \pm \sqrt{a^2 \pm x^2} + C$$