[高数]扩展后的基本积分公式列表

(1)

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

扩展：

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

(2)

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

上式中，由于不确定 $x$ 是否大于 $0$, 而在 $\ln x$ 中，$x$ 必须大于 $0$, 因此这里要加上绝对值。

扩展：

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

(3)

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

扩展：

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

(4)

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

(5)

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

(6)

$$ \int \sin x d x = – \cos x + C. $$

扩展：

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

(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)

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

扩展：

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

(19)

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

扩展：

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

(20)

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

EOF