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
\[
(\sqrt{\sin x})'=\frac{\cos x}{2\sqrt{\sin x}}.
\]
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 $x^2=\tan t$, $t\in(0,\frac{\pi}{2})$.
Posted by haifeng on 2016-04-28 17:54:53 last update 2016-04-28 19:31:44 | Answers (1) | 收藏
\[
\int\frac{1+\cos x}{1+\sin^2 x}dx
\]
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
\[
\int\frac{x+\sin x}{1+\cos x}dx.
\]
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.
Posted by haifeng on 2015-02-11 15:36:20 last update 2015-02-11 15:36:20 | Answers (0) | 收藏
求
\[\int e^{x^n}dx\]
Posted by haifeng on 2015-01-16 21:59:32 last update 2020-11-17 10:50:26 | Answers (1) | 收藏
求不定积分
\[\int\sqrt{1+x^2}dx\]
[Hint] 碰到根号, 通常我们需要想办法将之去掉. 一个办法是令 $x=\tan t$, 另一个是令 $x=\sinh t$.
Posted by haifeng on 2015-01-12 14:11:16 last update 2015-01-12 14:11:16 | Answers (1) | 收藏
求不定积分
\[
\int\sqrt{\frac{e^x-1}{e^x+1}}dx.
\]
Posted by haifeng on 2014-12-23 20:25:34 last update 2022-04-26 08:56:59 | Answers (0) | 收藏
这里 $a > 0$.
\[
\int\frac{1}{t^2-a^2}\mathrm{d}t=\frac{1}{2a}\ln\biggl|\frac{t-a}{t+a}\biggr|+C,
\]
\[
\int\frac{1}{t^2+a^2}\mathrm{d}t=\frac{1}{a}\arctan\frac{t}{a}+C,
\]
Posted by haifeng on 2014-12-23 20:09:56 last update 2014-12-23 20:16:34 | Answers (2) | 收藏
求
\[\int\frac{x^2+1}{x^4+1}dx\]
用类似的方法, 求
\[\int\frac{x^2-1}{x^4+1}dx\]