Questions in category: 不定积分 (Indefinite Integral)

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## 1. 万能变换

Posted by haifeng on 2022-11-08 17:21:14 last update 2022-11-08 17:21:14 | Answers (0) | 收藏

\begin{aligned} \sin x=\frac{2t}{1+t^2},\\ \cos x=\frac{1-t^2}{1+t^2},\\ \tan x=\frac{2t}{1-t^2} \end{aligned}

## 2. 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 (2) | 收藏

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

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

## 3. 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

## 4. 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

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

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

## 5. 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

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

$\int\frac{\ln t}{t-2}\mathrm{d}t$

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

Posted by haifeng on 2021-11-20 15:47:53 last update 2023-03-18 08:19:15 | Answers (1) | 收藏

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

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

$\displaystyle\int\frac{4}{3+\cos 4x}\mathrm{d}x$

## 7. 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

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

[Hint] 令 $x=\frac{1}{t}$ 或 $x=\tan t$

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

## 9. 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 (1) | 收藏

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

References:

## 10. 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

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

Note that

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

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