# [高数]扩展后的基本积分公式列表

## 注意

• 以下公式中所有 $x$ 都可以整体替换成方块 $\square$，也就是说，下面公式中的 $x$ 可以替换成任意包含变量的式子，但要注意的是，要替换则整个式子中的 $x$ 都要统一替换。
• 用不定积分时不要忘记在式子的最后加上常数 $C$.

## 公式

(1)

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

$$\int \frac{1}{u^{2}}dx = -\frac{1}{u} + C.$$

(2)

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

$$\int \frac{-3}{x-2}dx = -3 \ln |x-2| + C.$$

(3)

$$\int a^{x}dx = \frac{a^{x}}{\ln a} + C.$$

$$\int a^{-x}dx = \frac{a^{-x}}{\ln a} + C.$$

(4)

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

(5)

$$\int \cos x dx = \sin x + C.$$

(6)

$$\int \sin x d x = – \cos x + C.$$

$$\int \sin k \pi x dx = -\frac{1}{k \pi} \cos k \pi x + C.$$

(7)

$$\int \frac{1}{\sin x}dx = \int \csc x dx = \ln |\csc x – \cot x| + C.$$

(8)

$$\int \frac{1}{\cos x}dx = \int \sec x dx = \ln |\sec x + \tan x| + C.$$

(9)

$$\int \frac{1}{\sin ^{2} x} dx = \int \csc ^{2} x dx = – \cot x + C.$$

(10)

$$\int \frac{1}{\cos ^{2} x} dx = \int \sec^{2}x dx = \tan x + C.$$

(11)

$$\int \tan x dx = – \ln |\cos x| + C.$$

(12)

$$\int \cot x dx = \ln |\sin x| + C.$$

(13)

$$\int \sec x \tan x dx = \sec x +C.$$

(14)

$$\int \csc x \cot x dx = -\csc x +C.$$

(15)

$$\int \frac{1}{a^{2}+x^{2}}dx = \frac{1}{a} \arctan \frac{x}{a} + C.$$

(16)

$$\int \frac{1}{1+x^{2}}dx = \arctan x + C.$$

(17)

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

(18)

$$\int \frac{1}{\sqrt{a^{2} – x^{2}}}dx = \arcsin \frac{x}{a} + C.$$

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

(19)

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

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

(20)

$$\int \frac{1}{\sqrt{x^{2} \pm a^{2}}}dx = \ln \left | x+\sqrt{x^{2} \pm a^{2}} \right| + C.$$

EOF