# Quod Erat Demonstrandum

## 2017/03/19

### 盛水水深

Filed under: Additional / Applied Mathematics,HKCEE,NSS — johnmayhk @ 12:43 下午
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$(H-\sqrt[3]{H^3-h^3})$ 單位。

## 2016/06/06

### 和扇形有關的某積分

Filed under: mathematics,NSS — johnmayhk @ 12:27 下午
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$\displaystyle \int^1_0\sqrt{1-x^2}dx$

$\displaystyle \int^1_0\sqrt{1-x^2}dx=\frac{\pi}{4}$

$\displaystyle \int_0^t\sqrt{1-x^2}dx$　（其中 $0 < t < 1$

$x=\sin\theta$ 又或用部分積分吧。但以圖示之，求上述積分，即是求下圖著色部分面積：

(more…)

## 2015/11/02

### 一個冇咩用嘅rule

Filed under: NSS,Pure Mathematics — johnmayhk @ 1:53 下午
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$u=u(x)$$v=v(x)$ 二次可導，若 $\frac{d^2u}{dx^2}=au$$\frac{d^2v}{dx^2}=bv$ 其中 $a,b$ 為常數；則

$\int uvdx=\frac{1}{b-a}(u\frac{dv}{dx}-v\frac{du}{dx})+C$

## 2015/08/09

### 某積分

Filed under: NSS — johnmayhk @ 12:06 上午
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$\int x^2\sqrt{1+x^2}dx$

$x=\tan\theta$

$\int\sec^3\theta d\theta$$\int\sec^5\theta d\theta$

## 2015/05/14

### 某積分

Filed under: NSS — johnmayhk @ 5:37 下午
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$\displaystyle \int\frac{e^x(x\ln x+1)}{x}dx$

## 2015/02/20

### 鈄截柱體體積

Filed under: Junior Form Mathematics,NSS,Physics — johnmayhk @ 8:25 下午
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1.緣起

$V=A\frac{h_1+h_2+h_3}{3}$

$V=A\frac{0+0+h}{3}=\frac{1}{3}Ah$ (more…)

## 2014/08/11

### 吃驚積分

Filed under: Fun,Pure Mathematics — johnmayhk @ 9:18 下午
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$\displaystyle \int_0^\infty \frac{\sin x}{x}dx=\frac{\pi}{2}$

http://www.wolframalpha.com/input/?i=integrate+sin%28x%29%2Fx+from+0+to+infinity

$\displaystyle \int_0^\infty \frac{\sin x}{x}\cdot \frac{\sin (x/3)}{(x/3)}dx=\frac{\pi}{2}$

http://www.wolframalpha.com/input/?i=integrate+%28sin%28x%29%2Fx%29%28sin%28x%2F3%29%2F%28x%2F3%29%29+from+0+to+infinity

## 2014/08/08

### 利用比較係數做積分

Filed under: NSS,Pure Mathematics — johnmayhk @ 12:16 上午
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https://johnmayhk.wordpress.com/2012/02/07/integration-by-parts/

$\frac{d}{dx}P(x)e^x=(P(x)+P'(x))e^x$

$P(x)$$P(x)+P'(x)$ 皆是多項式，且它們的次數（degree）相同，

## 2014/06/16

### 積分二三事

Filed under: NSS,Pure Mathematics — johnmayhk @ 11:50 下午
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1.

$\int \frac{dx}{1-x^2}$

$x=\sin \theta$（或 $x=\cos \theta$

## 2014/06/11

### M2 某題

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 9:06 上午
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M2 學生問以下一題

The slope at any point (x,y) of a curve is given by

$\frac{dy}{dx}=y(2x+9)$.

If the curve lies above the x-axis, and it passes (0,8), find the equation of the curve.
(more…)

## 2011/10/10

### 錯在哪裡之 0 = 1

Filed under: HKALE,NSS,Pure Mathematics — johnmayhk @ 5:29 上午
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Old stuff…

$\int \frac{\cos \theta d\theta}{\sin \theta}$

$=\int \frac{d\sin \theta}{\sin \theta}$ (more…)

## 2011/05/22

### 某關於積分的純數題

Filed under: Pure Mathematics — johnmayhk @ 6:34 下午
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$\displaystyle\lim_{x\rightarrow +\infty}f(x)$

$\displaystyle\lim_{x\rightarrow +\infty}\int_{a}^{x}f(t)dt$ (其中常數 $a>0$)

## 2010/01/06

### 錯在哪裡：代入法

Filed under: HKALE,Pure Mathematics — johnmayhk @ 5:17 下午
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（2001-AL-Pure Mathematics-II-Q.12(a)(ii)）

$\int \frac{x^2 + 1}{(x^2 - x + 1)(x^2 + x + 1)}dx = \frac{1}{2} \int \frac{dx}{x^2 - x + 1} + \frac{1}{2} \int \frac{dx}{x^2 + x + 1}$

$I$
$= \frac{1}{2} \int \frac{-dy}{y^2 - y + 1}$
$= \frac{1}{2} \int \frac{-dx}{x^2 - x + 1}$ (dummy variable)