常用积分表 - 老官童鞋gogo的博客

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常用积分表

2025-06-12

数学

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高等数学

微积分

一、常见积分表#

序号	积分表达式	积分结果
1	$\int x^n \, \mathrm{d}x$	$\dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
2	$\int \mathrm{d}x$	$x + C$
3	$\int a \, \mathrm{d}x$	$a x + C$
4	$\int \frac{1}{x} \, \mathrm{d}x$	$\ln\vert x\vert + C$
5	$\int e^{x} \, \mathrm{d}x$	$e^{x} + C$
6	$\int a^{x} \, \mathrm{d}x$	$\dfrac{a^{x}}{\ln a} + C \quad (a>0, a\neq 1)$
7	$\int \sin x \, \mathrm{d}x$	$-\cos x + C$
8	$\int \cos x \, \mathrm{d}x$	$\sin x + C$
9	$\int \tan x \, \mathrm{d}x$	$-\ln\vert \cos x\vert + C$
10	$\int \cot x \, \mathrm{d}x$	$\ln\vert \sin x\vert + C$
11	$\int \sec x \, \mathrm{d}x$	$\ln\vert \sec x + \tan x\vert + C$
12	$\int \csc x \, \mathrm{d}x$	$\ln\vert \csc x - \cot x\vert + C$
13	$\int \sec^2 x \, \mathrm{d}x$	$\tan x + C$
14	$\int \csc^2 x \, \mathrm{d}x$	$-\cot x + C$
15	$\int \sec x \tan x \, \mathrm{d}x$	$\sec x + C$
16	$\int \csc x \cot x \, \mathrm{d}x$	$-\csc x + C$
17	$\int \frac{1}{1 + x^2} \, \mathrm{d}x$	$\arctan x + C$
18	$\int \frac{1}{\sqrt{1 - x^2}} \, \mathrm{d}x$	$\arcsin x + C$
19	$\int \frac{1}{\sqrt{x^2 - 1}} \, \mathrm{d}x$	$\ln\vert x + \sqrt{x^2 - 1}\vert + C$
20	$\int \ln x \, \mathrm{d}x$	$x \ln x - x + C$
21	$\int x e^{x} \, \mathrm{d}x$	$(x-1)e^x + C$
22	$\int x \sin x \, \mathrm{d}x$	$-\!x \cos x + \sin x + C$
23	$\int x \cos x \, \mathrm{d}x$	$x \sin x + \cos x + C$
24	$\int e^{a x} \, \mathrm{d}x$	$\frac{1}{a} e^{a x} + C$
25	$\int \frac{1}{a^2 + x^2} \, \mathrm{d}x$	$\frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$
26	$\int \frac{1}{\sqrt{a^2 - x^2}} \, \mathrm{d}x$	$\arcsin\left(\frac{x}{a}\right) + C$
27	$\int \frac{1}{x^2 - a^2} \, \mathrm{d}x$	$\frac{1}{2a} \ln\left\vert \frac{x-a}{x+a}\right\vert + C$
28	$\int \frac{1}{\vert x\vert } \, \mathrm{d}x$	$\operatorname{sgn}(x)\ln\vert x\vert + C$
29	$\int \arctan x \, \mathrm{d}x$	$x \arctan x - \frac{1}{2} \ln(1 + x^2) + C$
30	$\int \arcsin x \, \mathrm{d}x$	$x \arcsin x + \sqrt{1 - x^2} + C$

二、详细证明#

下面对上述表格中的部分积分公式进行详细推导。其余积分的推导可类似进行。

1. 幂函数积分#

\int x^n \, \mathrm{d}x = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

证明：

对 $x^n$ 积分，设 $F(x) = \frac{x^{n+1}}{n+1}$ ，则

\frac{\mathrm{d}}{\mathrm{d}x} F(x) = \frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{x^{n+1}}{n+1} \right) = x^n

因此，

\int x^n \, \mathrm{d}x = \frac{x^{n+1}}{n+1} + C

2. $\int \mathrm{d}x = x + C$ #

证明：

\frac{\mathrm{d}}{\mathrm{d}x} x = 1

故积分结果为 $x + C$ 。

3. $\int a \, \mathrm{d}x = a x + C$ #

证明：

\frac{\mathrm{d}}{\mathrm{d}x} (a x) = a

故积分结果为 $a x + C$ 。

4. $\int \frac{1}{x} \mathrm{d}x$ #

\int \frac{1}{x} \, \mathrm{d}x = \ln|x| + C

证明：

$\frac{\mathrm{d}}{\mathrm{d}x}\ln|x| = \frac{1}{x}$ ，故直接得证。

5. $\int e^x \, \mathrm{d}x$ #

\int e^{x} \, \mathrm{d}x = e^{x} + C

证明：

$\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$ ，故直接得证。

6. $\int a^x \, \mathrm{d}x$ #

\int a^{x} \, \mathrm{d}x = \frac{a^{x}}{\ln a} + C, \quad (a > 0, a \neq 1)

证明：

$\frac{\mathrm{d}}{\mathrm{d}x} a^x = a^x \ln a$ ，于是

\frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{a^x}{\ln a} \right) = a^x

7. $\int \sin x \, \mathrm{d}x$ #

\int \sin x \, \mathrm{d}x = -\cos x + C

证明：

$\frac{\mathrm{d}}{\mathrm{d}x}(-\cos x) = \sin x$

8. $\int \cos x \, \mathrm{d}x$ #

\int \cos x \, \mathrm{d}x = \sin x + C

证明：

$\frac{\mathrm{d}}{\mathrm{d}x}(\sin x) = \cos x$

9. $\int \tan x \, \mathrm{d}x$ #

\int \tan x \, \mathrm{d}x = -\ln|\cos x| + C

证明：

$\tan x = \frac{\sin x}{\cos x}$

令 $u = \cos x$ ， $\mathrm{d}u = -\sin x \mathrm{d}x \implies -\mathrm{d}u = \sin x \mathrm{d}x$

\int \tan x \, \mathrm{d}x = \int \frac{\sin x}{\cos x} \mathrm{d}x

令 $u = \cos x$ ，则 $\mathrm{d}u = -\sin x \mathrm{d}x$ ，所以

\int \tan x \, \mathrm{d}x = -\int \frac{1}{u} \mathrm{d}u = -\ln|u| + C = -\ln|\cos x| + C

10. $\int \cot x \, \mathrm{d}x = \ln|\sin x| + C$ #

证明：

$\cot x = \frac{\cos x}{\sin x}$ ，令 $u = \sin x$ ， $\mathrm{d}u = \cos x \mathrm{d}x$ ，则

\int \cot x \, \mathrm{d}x = \int \frac{\cos x}{\sin x} \mathrm{d}x = \int \frac{1}{u} \mathrm{d}u = \ln|u| + C = \ln|\sin x| + C

11. $\int \sec x \, \mathrm{d}x = \ln|\sec x + \tan x| + C$ #

证明：

考虑分子分母同乘 $\sec x + \tan x$ ：

\int \sec x \mathrm{d}x = \int \frac{\sec x (\sec x + \tan x)}{\sec x + \tan x} \mathrm{d}x = \int \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} \mathrm{d}x

令 $u = \sec x + \tan x$ ，则

\frac{\mathrm{d}u}{\mathrm{d}x} = \sec x \tan x + \sec^2 x = \sec^2 x + \sec x \tan x

所以

\int \sec x \mathrm{d}x = \int \frac{1}{u} \mathrm{d}u = \ln|u| + C = \ln|\sec x + \tan x| + C

12. $\int \csc x \, \mathrm{d}x = \ln|\csc x - \cot x| + C$ #

证明：

分子分母同乘 $\csc x - \cot x$ ：

\int \csc x \mathrm{d}x = \int \frac{\csc x (\csc x - \cot x)}{\csc x - \cot x} \mathrm{d}x = \int \frac{\csc^2 x - \csc x \cot x}{\csc x - \cot x} \mathrm{d}x

令 $u = \csc x - \cot x$ ，则

\frac{\mathrm{d}u}{\mathrm{d}x} = -\csc x \cot x + \csc^2 x = \csc^2 x - \csc x \cot x

所以

\int \csc x \mathrm{d}x = \int \frac{1}{u} \mathrm{d}u = \ln|u| + C = \ln|\csc x - \cot x| + C

13. $\int \sec^2 x \, \mathrm{d}x = \tan x + C$ #

证明：

\frac{\mathrm{d}}{\mathrm{d}x} \tan x = \sec^2 x

14. $\int \csc^2 x \, \mathrm{d}x = -\cot x + C$ #

证明：

\frac{\mathrm{d}}{\mathrm{d}x} (-\cot x) = \csc^2 x

15. $\int \sec x \tan x \, \mathrm{d}x = \sec x + C$ #

证明：

\frac{\mathrm{d}}{\mathrm{d}x} \sec x = \sec x \tan x

16. $\int \csc x \cot x \, \mathrm{d}x = -\csc x + C$ #

证明：

\frac{\mathrm{d}}{\mathrm{d}x} (-\csc x) = \csc x \cot x

17. $\int \frac{1}{1+x^2}\mathrm{d}x$ #

\int \frac{1}{1 + x^2} \, \mathrm{d}x = \arctan x + C

证明：

$\frac{\mathrm{d}}{\mathrm{d}x} \arctan x = \frac{1}{1 + x^2}$

18. $\int \frac{1}{\sqrt{1-x^2}}\mathrm{d}x$ #

\int \frac{1}{\sqrt{1 - x^2}} \, \mathrm{d}x = \arcsin x + C

证明：

$\frac{\mathrm{d}}{\mathrm{d}x} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$

19. $\int \frac{1}{\sqrt{x^2 - 1}} \, \mathrm{d}x = \ln|x + \sqrt{x^2 - 1}| + C$ #

证明：

令 $x = \cosh t$ ， $\mathrm{d}x = \sinh t \mathrm{d}t$ ， $\sqrt{x^2 - 1} = \sinh t$ 。

\int \frac{1}{\sqrt{x^2 - 1}} \mathrm{d}x = \int \frac{1}{\sinh t} \sinh t \mathrm{d}t = \int 1 \mathrm{d}t = t + C

又 $x = \cosh t \implies t = \operatorname{arcosh} x = \ln|x + \sqrt{x^2 - 1}|$

\int \frac{1}{\sqrt{x^2 - 1}} \mathrm{d}x = \ln|x + \sqrt{x^2 - 1}| + C

20. $\int \ln x \, \mathrm{d}x$ #

分部积分法，令 $u = \ln x$ ， $\mathrm{d}v = \mathrm{d}x$ ，则 $\mathrm{d}u = \frac{1}{x} \mathrm{d}x$ ， $v = x$

\int \ln x \, \mathrm{d}x = x \ln x - \int x \cdot \frac{1}{x} \mathrm{d}x = x \ln x - \int 1 \, \mathrm{d}x = x \ln x - x + C

21. $\int x e^x \, \mathrm{d}x = (x-1) e^x + C$ #

证明：

分部积分， $u = x$ , $\mathrm{d}v = e^x \mathrm{d}x$ , $\mathrm{d}u = \mathrm{d}x$ , $v = e^x$

\int x e^x \mathrm{d}x = x e^x - \int e^x \mathrm{d}x = x e^x - e^x + C = (x-1) e^x + C

22. $\int x \sin x \, \mathrm{d}x$ #

分部积分法，令 $u = x$ ， $\mathrm{d}v = \sin x \, \mathrm{d}x$ ，则 $\mathrm{d}u = \mathrm{d}x$ ， $v = -\cos x$

\int x \sin x \, \mathrm{d}x = -x \cos x + \int \cos x \, \mathrm{d}x = -x \cos x + \sin x + C

23. $\int x \cos x \, \mathrm{d}x$ #

分部积分法，令 $u = x$ ， $\mathrm{d}v = \cos x \, \mathrm{d}x$ ，则 $\mathrm{d}u = \mathrm{d}x$ ， $v = \sin x$

\int x \cos x \, \mathrm{d}x = x \sin x - \int \sin x \, \mathrm{d}x = x \sin x + \cos x + C

24. $\int e^{a x} \, \mathrm{d}x = \frac{1}{a} e^{a x} + C$ #

证明：

$\frac{\mathrm{d}}{\mathrm{d}x} e^{a x} = a e^{a x}$ ，所以

\int e^{a x} \mathrm{d}x = \frac{1}{a} e^{a x} + C

25. $\int \frac{1}{a^2 + x^2} \, \mathrm{d}x$ #

令 $x = a \tan \theta$ ，则 $\mathrm{d}x = a \sec^2 \theta \mathrm{d}\theta$ ， $a^2 + x^2 = a^2 + a^2 \tan^2 \theta = a^2 \sec^2 \theta$

\int \frac{1}{a^2 + x^2} \mathrm{d}x = \int \frac{1}{a^2 \sec^2 \theta} a \sec^2 \theta \mathrm{d}\theta = \int \frac{1}{a} \mathrm{d}\theta = \frac{1}{a} \theta + C

又 $x = a \tan \theta \implies \theta = \arctan\left(\frac{x}{a}\right)$

\int \frac{1}{a^2 + x^2} \mathrm{d}x = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C

26. $\int \frac{1}{\sqrt{a^2 - x^2}} \mathrm{d}x$ #

令 $x = a \sin \theta$ ， $\mathrm{d}x = a \cos \theta \mathrm{d}\theta$ ， $\sqrt{a^2 - x^2} = a \cos \theta$

\int \frac{1}{\sqrt{a^2 - x^2}} \mathrm{d}x = \int \frac{1}{a \cos \theta} a \cos \theta \mathrm{d}\theta = \int 1 \mathrm{d}\theta = \theta + C

又 $x = a \sin \theta \implies \theta = \arcsin \left(\frac{x}{a}\right)$

\int \frac{1}{\sqrt{a^2 - x^2}} \mathrm{d}x = \arcsin \left(\frac{x}{a}\right) + C

27. $\int \frac{1}{x^2 - a^2} \mathrm{d}x$ #

可拆分为部分分式：

\frac{1}{x^2 - a^2} = \frac{1}{2a}\left( \frac{1}{x-a} - \frac{1}{x+a} \right)

所以

\int \frac{1}{x^2 - a^2} \mathrm{d}x = \frac{1}{2a} \int \left( \frac{1}{x-a} - \frac{1}{x+a} \right) \mathrm{d}x = \frac{1}{2a} \ln \left| \frac{x-a}{x+a} \right| + C

28. $\int \frac{1}{|x|} \mathrm{d}x = \operatorname{sgn}(x) \ln|x| + C$ #

证明：

$\frac{1}{|x|} = \frac{1}{x}$ (当 $x>0$ ), $= -\frac{1}{x}$ (当 $x<0$ ) 所以

\int \frac{1}{|x|} \mathrm{d}x = \begin{cases} \ln x + C, & x>0 \\ -\ln(-x) + C, & x < 0 \end{cases}

合并表示为 $\operatorname{sgn}(x)\ln|x| + C$

29. $\int \arctan x \, \mathrm{d}x = x \arctan x - \frac{1}{2} \ln(1+x^2) + C$ #

证明： 分部积分， $u = \arctan x$ , $\mathrm{d}v = \mathrm{d}x$ , $\mathrm{d}u = \frac{1}{1+x^2} \mathrm{d}x$ , $v = x$

\int \arctan x \mathrm{d}x = x \arctan x - \int \frac{x}{1+x^2} \mathrm{d}x

令 $w = 1 + x^2$ , $\mathrm{d}w = 2x \mathrm{d}x$

\int \frac{x}{1+x^2} \mathrm{d}x = \frac{1}{2} \int \frac{1}{w} \mathrm{d}w = \frac{1}{2} \ln|w| + C = \frac{1}{2} \ln(1+x^2)

所以

\int \arctan x \mathrm{d}x = x \arctan x - \frac{1}{2} \ln(1+x^2) + C

30. $\int \arcsin x \, \mathrm{d}x = x \arcsin x + \sqrt{1-x^2} + C$ #

证明：

分部积分， $u = \arcsin x$ , $\mathrm{d}v = \mathrm{d}x$ , $\mathrm{d}u = \frac{1}{\sqrt{1-x^2}} \mathrm{d}x$ , $v = x$

\int \arcsin x \mathrm{d}x = x \arcsin x - \int x \cdot \frac{1}{\sqrt{1-x^2}} \mathrm{d}x

对于 $\int x (1-x^2)^{-1/2} \mathrm{d}x$ ，令 $w = 1-x^2$ , $\mathrm{d}w = -2x \mathrm{d}x \implies -\frac{1}{2} \mathrm{d}w = x \mathrm{d}x$

\int x (1-x^2)^{-1/2} \mathrm{d}x = -\frac{1}{2} \int w^{-1/2} \mathrm{d}w = -\frac{1}{2} \cdot 2 w^{1/2} = - \sqrt{1-x^2}

所以

\int \arcsin x \mathrm{d}x = x \arcsin x + \sqrt{1-x^2} + C

常用积分表

https://www.laoguantx.cn/posts/commonintegraltable/

作者

老官童鞋gogo

发布于

2025-06-12

许可协议

CC BY-NC-SA 4.0

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最小二乘法

常微分方程求解（6）

一、常见积分表#

二、详细证明#

1. 幂函数积分#

2. ∫dx=x+C\int \mathrm{d}x = x + C∫dx=x+C#

3. ∫a dx=ax+C\int a \, \mathrm{d}x = a x + C∫adx=ax+C#

4. ∫1xdx\int \frac{1}{x} \mathrm{d}x∫x1​dx#

5. ∫ex dx\int e^x \, \mathrm{d}x∫exdx#

6. ∫ax dx\int a^x \, \mathrm{d}x∫axdx#

7. ∫sin⁡x dx\int \sin x \, \mathrm{d}x∫sinxdx#

8. ∫cos⁡x dx\int \cos x \, \mathrm{d}x∫cosxdx#

9. ∫tan⁡x dx\int \tan x \, \mathrm{d}x∫tanxdx#

10. ∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x \, \mathrm{d}x = \ln|\sin x| + C∫cotxdx=ln∣sinx∣+C#

11. ∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+C\int \sec x \, \mathrm{d}x = \ln|\sec x + \tan x| + C∫secxdx=ln∣secx+tanx∣+C#

12. ∫csc⁡x dx=ln⁡∣csc⁡x−cot⁡x∣+C\int \csc x \, \mathrm{d}x = \ln|\csc x - \cot x| + C∫cscxdx=ln∣cscx−cotx∣+C#

13. ∫sec⁡2x dx=tan⁡x+C\int \sec^2 x \, \mathrm{d}x = \tan x + C∫sec2xdx=tanx+C#

14. ∫csc⁡2x dx=−cot⁡x+C\int \csc^2 x \, \mathrm{d}x = -\cot x + C∫csc2xdx=−cotx+C#

15. ∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec x \tan x \, \mathrm{d}x = \sec x + C∫secxtanxdx=secx+C#

16. ∫csc⁡xcot⁡x dx=−csc⁡x+C\int \csc x \cot x \, \mathrm{d}x = -\csc x + C∫cscxcotxdx=−cscx+C#

17. ∫11+x2dx\int \frac{1}{1+x^2}\mathrm{d}x∫1+x21​dx#

18. ∫11−x2dx\int \frac{1}{\sqrt{1-x^2}}\mathrm{d}x∫1−x2​1​dx#

19. ∫1x2−1 dx=ln⁡∣x+x2−1∣+C\int \frac{1}{\sqrt{x^2 - 1}} \, \mathrm{d}x = \ln|x + \sqrt{x^2 - 1}| + C∫x2−1​1​dx=ln∣x+x2−1​∣+C#

20. ∫ln⁡x dx\int \ln x \, \mathrm{d}x∫lnxdx#

21. ∫xex dx=(x−1)ex+C\int x e^x \, \mathrm{d}x = (x-1) e^x + C∫xexdx=(x−1)ex+C#

22. ∫xsin⁡x dx\int x \sin x \, \mathrm{d}x∫xsinxdx#

23. ∫xcos⁡x dx\int x \cos x \, \mathrm{d}x∫xcosxdx#

24. ∫eax dx=1aeax+C\int e^{a x} \, \mathrm{d}x = \frac{1}{a} e^{a x} + C∫eaxdx=a1​eax+C#

25. ∫1a2+x2 dx\int \frac{1}{a^2 + x^2} \, \mathrm{d}x∫a2+x21​dx#

26. ∫1a2−x2dx\int \frac{1}{\sqrt{a^2 - x^2}} \mathrm{d}x∫a2−x2​1​dx#

27. ∫1x2−a2dx\int \frac{1}{x^2 - a^2} \mathrm{d}x∫x2−a21​dx#

28. ∫1∣x∣dx=sgn⁡(x)ln⁡∣x∣+C\int \frac{1}{|x|} \mathrm{d}x = \operatorname{sgn}(x) \ln|x| + C∫∣x∣1​dx=sgn(x)ln∣x∣+C#

29. ∫arctan⁡x dx=xarctan⁡x−12ln⁡(1+x2)+C\int \arctan x \, \mathrm{d}x = x \arctan x - \frac{1}{2} \ln(1+x^2) + C∫arctanxdx=xarctanx−21​ln(1+x2)+C#

30. ∫arcsin⁡x dx=xarcsin⁡x+1−x2+C\int \arcsin x \, \mathrm{d}x = x \arcsin x + \sqrt{1-x^2} + C∫arcsinxdx=xarcsinx+1−x2​+C#

2. $\int \mathrm{d}x = x + C$ #

3. $\int a \, \mathrm{d}x = a x + C$ #

4. $\int \frac{1}{x} \mathrm{d}x$ #

5. $\int e^x \, \mathrm{d}x$ #

6. $\int a^x \, \mathrm{d}x$ #

7. $\int \sin x \, \mathrm{d}x$ #

8. $\int \cos x \, \mathrm{d}x$ #

9. $\int \tan x \, \mathrm{d}x$ #

10. $\int \cot x \, \mathrm{d}x = \ln|\sin x| + C$ #

11. $\int \sec x \, \mathrm{d}x = \ln|\sec x + \tan x| + C$ #

12. $\int \csc x \, \mathrm{d}x = \ln|\csc x - \cot x| + C$ #

13. $\int \sec^2 x \, \mathrm{d}x = \tan x + C$ #

14. $\int \csc^2 x \, \mathrm{d}x = -\cot x + C$ #

15. $\int \sec x \tan x \, \mathrm{d}x = \sec x + C$ #

16. $\int \csc x \cot x \, \mathrm{d}x = -\csc x + C$ #

17. $\int \frac{1}{1+x^2}\mathrm{d}x$ #

18. $\int \frac{1}{\sqrt{1-x^2}}\mathrm{d}x$ #

19. $\int \frac{1}{\sqrt{x^2 - 1}} \, \mathrm{d}x = \ln|x + \sqrt{x^2 - 1}| + C$ #

20. $\int \ln x \, \mathrm{d}x$ #

21. $\int x e^x \, \mathrm{d}x = (x-1) e^x + C$ #

22. $\int x \sin x \, \mathrm{d}x$ #

23. $\int x \cos x \, \mathrm{d}x$ #

24. $\int e^{a x} \, \mathrm{d}x = \frac{1}{a} e^{a x} + C$ #

25. $\int \frac{1}{a^2 + x^2} \, \mathrm{d}x$ #

26. $\int \frac{1}{\sqrt{a^2 - x^2}} \mathrm{d}x$ #

27. $\int \frac{1}{x^2 - a^2} \mathrm{d}x$ #

28. $\int \frac{1}{|x|} \mathrm{d}x = \operatorname{sgn}(x) \ln|x| + C$ #

29. $\int \arctan x \, \mathrm{d}x = x \arctan x - \frac{1}{2} \ln(1+x^2) + C$ #

30. $\int \arcsin x \, \mathrm{d}x = x \arcsin x + \sqrt{1-x^2} + C$ #