Questions in category: 不定积分 (Indefinite Integral)
分析 >> 数学分析 >> 不定积分
<[1] [2] [3] >

1. 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

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

2. 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

3. 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 2017-12-29 11:38:45 | Answers (0) | 收藏


计算不定积分

\[
\int\sqrt{\frac{1+x}{1-x}}dx
\]
 

 

Remark:

相关题目参见: 问题2047

4. 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 2017-12-29 11:22:53 | Answers (0) | 收藏


证明:

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

5. 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 

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

6. 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 $x^2=\tan t$,   $t\in(0,\frac{\pi}{2})$.

7. 求不定积分 $\int\frac{1+\cos x}{1+\sin^2 x}dx$.

Posted by haifeng on 2016-04-28 17:54:53 last update 2016-04-28 19:31:44 | Answers (1) | 收藏


\[
\int\frac{1+\cos x}{1+\sin^2 x}dx
\]

 

8. 求不定积分 $\int\frac{x+\sin x}{1+\cos x}dx$.

Posted by haifeng on 2016-02-20 00:50:48 last update 2016-02-20 00:50:48 | Answers (1) | 收藏


求不定积分

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

9. $\int(1+x-\frac{1}{x})e^{x+\frac{1}{x}}dx$

Posted by haifeng on 2015-03-03 10:44:53 last update 2015-03-03 13:31:43 | Answers (0) | 收藏


\[
\int(1+x-\frac{1}{x})e^{x+\frac{1}{x}}dx
\]


[Hint]

考虑 $de^{x+\frac{1}{x}}$, 然后可以迅速找到被积函数的原函数.

10. 求 $\int e^{x^n}dx$

Posted by haifeng on 2015-02-11 15:36:20 last update 2015-02-11 15:36:20 | Answers (0) | 收藏


\[\int e^{x^n}dx\]

 

<[1] [2] [3] >