不定积分表 ¶
背景补充 ¶
双曲正弦函数 ¶
\(y = \sinh x = \dfrac{e^x - e^{-x}}{2}\)
反函数
\(y = \ln(x + \sqrt{x^2 + 1})\)
双曲余弦函数 ¶
\(y = \cosh x = \dfrac{e^x + e^{-x}}{2}\)
反函数
\(y = \ln(x + \sqrt{x^2 - 1})\)
双曲正切函数 ¶
\(y = \tanh x = \dfrac{\sinh x}{\cosh x} = \dfrac{e^x - e^{-x}}{e^x + e^{-x}}\)
反函数
\(y = \dfrac{1}{2}\ln\dfrac{1 + x}{1 - x}\)
为何叫双曲三角函数呢，是因为他们跟三角函数有类似的性质，甚至存在双曲三角函数线，有兴趣的同学可以参考知乎的两篇文章
了解了双曲三角函数以后，一些不定积分公式也容易理解了
1.\(\displaystyle\int k \, \mathrm{d}x = kx + C\)
2.\(\displaystyle\int x^\alpha \, \mathrm{d}x = \frac{x^{\alpha+1}}{\alpha+1} + C \quad (\alpha \neq -1)\)
3.\(\displaystyle\int \frac{1}{x} \, \mathrm{d}x = \ln|x| + C\)
4.\(\displaystyle\int \frac{\mathrm{d}x}{1+x^2} = \arctan x + C\)
5.\(\displaystyle\int \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \arcsin x + C\)
6.\(\displaystyle\int \cos x \, \mathrm{d}x = \sin x + C\)
7.\(\displaystyle\int \sin x \, \mathrm{d}x = -\cos x + C\)
8.\(\displaystyle\int \frac{\mathrm{d}x}{\cos^2 x} = \tan x + C\)
9.\(\displaystyle\int \frac{\mathrm{d}x}{\sin^2 x} = -\cot x + C\)
10.\(\displaystyle\int \sec x \tan x \, \mathrm{d}x = \sec x + C\)
11.\(\displaystyle\int \csc x \cot x \, \mathrm{d}x = -\csc x + C\)
12.\(\displaystyle\int a^x \, \mathrm{d}x = \frac{a^x}{\ln a} + C\)
13.\(\displaystyle\int \sinh x \, \mathrm{d}x = \cosh x + C\)
14.\(\displaystyle\int \cosh x \, \mathrm{d}x = \sinh x + C\)
15.\(\displaystyle\int \tan x \, \mathrm{d}x = -\ln|\cos x| + C\)
16.\(\displaystyle\int \cot x \, \mathrm{d}x = \ln|\sin x| + C\)
17.\(\displaystyle\int \sec x \, \mathrm{d}x = \ln|\sec x + \tan x| + C\)
18.\(\displaystyle\int \csc x \, \mathrm{d}x = \ln|\csc x - \cot x| + C\)
19.\(\displaystyle\int \frac{\mathrm{d}x}{a^2+x^2} = \frac{1}{a} \arctan \frac{x}{a} + C\)
20.\(\displaystyle\int \frac{\mathrm{d}x}{a^2-x^2} = \frac{1}{2a} \ln \left| \frac{x+a}{a-x} \right| + C\)
21.\(\displaystyle\int \frac{\mathrm{d}x}{\sqrt{a^2-x^2}} = \arcsin \frac{x}{a} + C\)
22.\(\displaystyle\int \frac{\mathrm{d}x}{\sqrt{a^2+x^2}} = \ln \left( x + \sqrt{x^2+a^2} \right) + C\)
23.\(\displaystyle\int \frac{\mathrm{d}x}{\sqrt{x^2-a^2}} = \ln \left| x + \sqrt{x^2-a^2} \right| + C\)
24.\(\displaystyle\int \sqrt{a^2-x^2} \, \mathrm{d}x = \frac{1}{2} \left( x\sqrt{a^2-x^2} + a^2 \arcsin \frac{x}{a} \right) + C\)
25.\(\displaystyle\int \sqrt{x^2 \pm a^2} \, \mathrm{d}x = \frac{1}{2} \left( x\sqrt{x^2 \pm a^2} \pm a^2 \ln \left| x \pm \sqrt{x^2 \pm a^2} \right| \right) + C\)
26.\(\displaystyle\int \frac{\mathrm{d}x}{(a^2+x^2)^2} = \frac{1}{2a^3} \left( \arctan \frac{x}{a} + \frac{ax}{x^2+a^2} \right) + C\)
\(\displaystyle\int \frac{\mathrm{d}x}{(1+x^2)^2} = \frac{1}{2} \left( \arctan x + \frac{x}{x^2+1} \right) + C\) \(\displaystyle\int \frac{x^2 \mathrm{d}x}{(1+x^2)^2} = \frac{1}{2} \left( \arctan x - \frac{x}{1+x^2} \right) + C\)
27.\(\displaystyle\int e^x \sin x \, \mathrm{d}x = \frac{1}{2} e^x (\sin x - \cos x) + C\)
28.\(\displaystyle\int e^x \cos x \, \mathrm{d}x = \frac{1}{2} e^x (\sin x + \cos x) + C\)