Questions in category: 不定积分 (Indefinite Integral)
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1. 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 $x=\tan t$

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

 

3. 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

 

4. 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

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

5. 计算不定积分 $\displaystyle\int\frac{2\sin x+\cos x}{\sin x+2\cos x}dx$

Posted by haifeng on 2017-12-29 11:41:45 last update 2017-12-29 11:41:45 | Answers (0) | 收藏


计算不定积分

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

6. 计算不定积分 $\displaystyle\int\frac{1}{1-\sin x}dx$

Posted by haifeng on 2017-12-29 11:40:25 last update 2018-08-11 22:13:48 | Answers (1) | 收藏


计算不定积分

\[
\int\frac{1}{1-\sin x}dx
\]

 

关于

\[
\int\frac{1+\sin x}{1-\sin x}dx
\]
 

参见问题 1450

7. 计算不定积分 $\displaystyle\int\sqrt{\frac{1+x}{1-x}}dx$

Posted by haifeng on 2017-12-29 11:36:45 last update 2020-12-07 10:54:53 | Answers (1) | 收藏


计算不定积分

\[
\int\sqrt{\frac{1+x}{1-x}}dx
\]
 

[Hint] 被积函数形如 $\sqrt[n]{\dfrac{ax+b}{cx+d}}$ 的可以整体令为 $t$.

 

Remark:

相关题目参见: 问题2047

8. 证明: $\arcsin x+\frac{\pi}{2}=2\arctan\sqrt{\frac{1+x}{1-x}}$

Posted by haifeng on 2017-12-29 11:22:53 last update 2020-12-07 11:21:55 | Answers (0) | 收藏


证明:

\[
\arcsin x+\frac{\pi}{2}=2\arctan\sqrt{\frac{1+x}{1-x}}
\]

 

相关问题参考问题2048.

9. 求不定积分 $\int\frac{e^{-\frac{x}{2}}(\cos x-\sin x)}{\sqrt{\sin x}}dx$.

Posted by haifeng on 2016-04-29 04:31:58 last update 2016-04-29 04:33:28 | Answers (1) | 收藏


\[
\int\frac{e^{-\frac{x}{2}}(\cos x-\sin x)}{\sqrt{\sin x}}dx
\]

 


[Hint] Note that 

\[
(\sqrt{\sin x})'=\frac{\cos x}{2\sqrt{\sin x}}.
\]

10. 求不定积分 $\int\frac{x^2+1}{x\sqrt{1+x^4}}dx$

Posted by haifeng on 2016-04-28 19:29:21 last update 2016-04-28 19:31:25 | Answers (1) | 收藏


求不定积分

\[
\int\frac{x^2+1}{x\sqrt{1+x^4}}dx
\]

 


[Hint] 令 $x^2=\tan t$,   $t\in(0,\frac{\pi}{2})$.

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