Questions in category: 不定积分 (Indefinite Integral)
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1. 求不定积分 $\displaystyle\int\arctan^2\theta\mathrm{d}\theta$.

Posted by haifeng on 2022-01-12 08:47:44 last update 2022-01-14 15:40:00 | Answers (1) | 收藏


求不定积分

\[\int\arctan^2\theta\mathrm{d}\theta=\int(\arctan\theta)^2\mathrm{d}\theta,\]

 

或写成 $x$ 的形式

\[\int(\arctan x)^2\mathrm{d}x.\]

 

2. 求不定积分 $\int\frac{\ln(1-\sqrt{1-x})}{x}\mathrm{d}x$.

Posted by haifeng on 2022-01-01 22:22:52 last update 2022-01-01 22:22:52 | Answers (1) | 收藏


求不定积分 $\int\frac{\ln(1-\sqrt{1-x})}{x}\mathrm{d}x$.

3. 求下列不定积分

Posted by haifeng on 2022-01-01 22:21:34 last update 2022-01-01 22:22:04 | Answers (0) | 收藏


\[
\int\frac{\ln(1-x)}{x}\mathrm{d}x,\qquad\int\frac{\ln(1+x)}{x}\mathrm{d}x
\]

 

\[
\int\frac{xe^x}{e^x-1}\mathrm{d}x,\qquad\int\frac{xe^x}{e^x+1}\mathrm{d}x
\]

4. 求不定积分 $\int\frac{x}{e^x-A}\mathrm{d}x$, 这里 $A$ 是一常数.

Posted by haifeng on 2021-12-29 23:06:14 last update 2021-12-29 23:23:10 | Answers (0) | 收藏


求不定积分 $\int\frac{x}{e^x-A}\mathrm{d}x$, 这里 $A$ 是一常数. 例如:

\[
\int\frac{x}{e^x-1}\mathrm{d}x,\quad \int\frac{x}{e^x-2}\mathrm{d}x,\quad \int\frac{x}{e^x+1}\mathrm{d}x
\]

等等.

 

 

计算不定积分

\[
\int\frac{\ln t}{t-2}\mathrm{d}t
\]

5. 求 $\int\frac{x^2}{1+\cos x}\mathrm{d}x$

Posted by haifeng on 2021-11-20 15:47:53 last update 2021-11-20 15:55:18 | Answers (0) | 收藏


求下列不定积分

$\displaystyle\int\frac{x^2}{1+\cos x}\mathrm{d}x$

 

$\displaystyle\int\frac{x^2}{\sin^2 x}\mathrm{d}x$

6. 求不定积分

Posted by haifeng on 2021-03-25 15:55:13 last update 2021-03-25 15:57:53 | Answers (1) | 收藏


求不定积分

\[
\int\frac{\mathrm{d}x}{x^2\sqrt{1+x^2}}
\]

 

 

[Hint] 令 $x=\frac{1}{t}$ 或 $x=\tan t$

7. [Homework] 4.1

Posted by haifeng on 2020-11-17 11:46:33 last update 2020-11-17 11:53:34 | Answers (3) | 收藏


P.167  习题 4.1


2. 求下列不定积分.

(6)    $\displaystyle\int(\sqrt{x}+1)(\sqrt{x^3}-1)dx$

 

 

(15)    $\displaystyle\int\frac{1}{1+\cos 2x}dx$

 

 

(20)    $\displaystyle\int\sqrt{1+\sin 2x}dx$,    $x\in(0,\frac{\pi}{2})$.

 

8. 计算 $\int\frac{1}{\sin^4(x)+\cos^4(x)}dx$.

Posted by haifeng on 2020-09-08 10:07:51 last update 2020-09-08 10:08:54 | Answers (0) | 收藏


计算

\[\int\frac{1}{\sin^4(x)+\cos^4(x)}dx\]

 

 

 


References:

 

对三大数学软件 Mathematica 、Maple 、MATLAB 的小测试比较

https://blog.csdn.net/iteye_18800/article/details/82294214?utm_medium=distribute.pc_relevant.none-task-blog-BlogCommendFromMachineLearnPai2-2.channel_param&depth_1-utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromMachineLearnPai2-2.channel_param

 

9. 求不定积分 $\int\frac{2(1-u)}{2u+u^2 e^{-u}}du$

Posted by haifeng on 2019-02-27 20:53:53 last update 2019-09-05 14:19:18 | Answers (0) | 收藏


求不定积分

\[\int\frac{2(1-u)}{2u+u^2 e^{-u}}du\]

 

Note that

\[
\frac{2(1-u)}{2u+u^2 e^{-u}}=\frac{1}{u}-\frac{e^{-u}+2}{ue^{-u}+2}
\]

10. 计算不定积分 $\displaystyle\int\frac{2\sin x+\cos x}{\sin x+2\cos x}\mathrm{d}x$

Posted by haifeng on 2017-12-29 11:41:45 last update 2021-11-14 11:28:21 | Answers (2) | 收藏


计算不定积分

\[
\int\frac{2\sin x+\cos x}{\sin x+2\cos x}\mathrm{d}x
\]
 

\[
\int\frac{\sin x}{\sin x+\cos x}\mathrm{d}x
\]

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