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Exponential and Logarithmic Integrals 指數和對數積分


使用指數法則 (Exponential Rule)


指數方程式的每一個微分法則都有相連結的積分法則


 -0.2cm指數方程式的積分

u 是一個 x 的微分方程式

\begin{displaymath}\begin{aligned}
\int e^x dx & = e^x + C & \hbox{ 簡單指數律 ...
...= \int e^u du = e^u + C & \hbox{ 一般指數律 } \\
\end{aligned}\end{displaymath}





範例 1    



積分指數方程式

(a)
$ \int$2exdx
(b)
$ \int$2e2xdx
(c)
$ \int$(ex + x)dx


解    

(a)
$ \int$2exdx = 2$ \int$ex = 2ex + C
(b)

$\displaystyle \int$2e2xdx = $\displaystyle \int$e(2x)(2)dx    
  = $\displaystyle \int$eu$\displaystyle {\frac{du}{dx}}$dx    
  = e2x + C    

(c)

$\displaystyle \int$(ex + x)dx = $\displaystyle \int$exdx + $\displaystyle \int$xdx    
  = ex + $\displaystyle {\frac{x^2}{2}}$ + C    

你可以用微分來檢視結果




範例 2    


積分一個指數方程式
尋找不定積分

$\displaystyle \int$e3x + 1dx


解     u = 3x + 1 使得 $ {\frac{du}{dx}}$ = 3

\begin{displaymath}\begin{aligned}
\int e^{3x + 1} & = \frac{1}{3} \int e^{3x +...
...} e^u + C \\
& = \frac{1}{3} e^{3x + 1} + C \\
\end{aligned}\end{displaymath}




範例 3    


積分一個指數方程式
尋找不定積分

$\displaystyle \int$5xe-x2dx


解    u = - x2 使得 $ {\frac{du}{dx}}$ = - 2x

\begin{displaymath}\begin{aligned}
\int 5xe^{-x^2} dx & = \int (- \frac{5}{2}) ...
...2} e^u + C \\
& = - \frac{5}{2} e^{-x^2} + C \\
\end{aligned}\end{displaymath}


使用對數律 (Log Rule)



在 5.1 和 5.2 節中提到積分的次方法則

\begin{displaymath}\begin{aligned}
\int x^n dx & = \frac{x^{n+1}}{n+1} + C , n ...
...{n+1} + C , n \neq -1 & \hbox{ 一般次方法則 } \\
\end{aligned}\end{displaymath}

對數法則讓你去積分 $ \int$x-1dx $ \int$u-1du


 -0.2cm對數方程式的積分

u 是一個 x 的可微分方程式

\begin{displaymath}\begin{aligned}
\int \frac{1}{x} dx & = \ln \vert x\vert + C...
...u = \ln \vert u\vert + C & \hbox{ 一般對數律} \\
\end{aligned}\end{displaymath}


你可以用微分來證明這些每一個法則,舉例,證明 d /dx[ln| x|] = 1/x 寫作

$\displaystyle {\frac{d}{dx}}$[ln x] = $\displaystyle {\frac{1}{x}}$        和        $\displaystyle {\frac{d}{dx}}$[ln(- x)] = $\displaystyle {\frac{-1}{-x}}$ = $\displaystyle {\frac{1}{x}}$




範例 4    


積分對數方程式

(a)
$ \int$$ {\frac{4}{x}}$dx
(b)
$ \int$$ {\frac{2x}{x^2}}$
(c)
$ \int$$ {\frac{3}{3x + 1}}$dx


解    

(a)

$\displaystyle \int$$\displaystyle {\frac{4}{x}}$dx = 4$\displaystyle \int$$\displaystyle {\frac{1}{x}}$dx    
  = 4 ln| x| + C    

(b)

$\displaystyle \int$$\displaystyle {\frac{2x}{x^2}}$ = $\displaystyle \int$$\displaystyle {\frac{du/dx}{u}}$dx    
  = ln| u| + C    
  = ln x2 + C    

(c)

$\displaystyle \int$$\displaystyle {\frac{3}{3x + 1}}$dx = $\displaystyle \int$$\displaystyle {\frac{du/dx}{u}}$dx    
  = ln| u| + C    
  = ln| 3x + 1| + C    




範例 5    


使用對數法則 (Log Rule)
尋找不定積分

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


解     u = 2x - 1 使得 du/dx = 2

\begin{displaymath}\begin{aligned}
\int \frac{1}{2x - 1} dx = \frac{1}{2} \int ...
...\\
& =\frac{1}{2} \ln \vert 2x - 1\vert + C \\
\end{aligned}\end{displaymath}




範例 6    


使用對數法則 (Log Rule)
尋找不定積分

$\displaystyle \int$$\displaystyle {\frac{6x}{x^2 + 1}}$dx


解     u = x2 + 1 使得 du/dx = 2x

\begin{displaymath}\begin{aligned}
\int \frac{6x}{x^2 + 1} dx & = 3 \int \frac{...
... \vert u\vert + C \\
& =3 \ln (x^2 + 1) + C \\
\end{aligned}\end{displaymath}




範例 7    


重複之前的積分
尋找不定積分

(a)
$\displaystyle \int$$\displaystyle {\frac{3x^3 + 2x - 1}{x^2}}$dx
(b)
$\displaystyle \int$$\displaystyle {\frac{1}{1 + e^{-x}}}$dx
(c)
$\displaystyle \int$$\displaystyle {\frac{x^2 + x + 1}{x - 1}}$dx


解    

(a)
先化成三個式子相加的積分

\begin{displaymath}\begin{aligned}
\int \frac{3x^2 + 2x - 1}{x^2} dx& = \int (\f...
...\
& =3x +2\ln \vert n\vert + \frac{1}{x} +C \\
\end{aligned}\end{displaymath}

(b)
先將原來的式子乘上 $ {\frac{e^x}{e^x}}$

\begin{displaymath}\begin{aligned}
\int \frac{1}{1 + e^{-x}}& = \int (\frac{e^x}...
...rac{e^x}{e^x + 1} dx\\
& = \ln (e^x + 1) + C\\
\end{aligned}\end{displaymath}

(c)
分子先除分母

\begin{displaymath}\begin{aligned}
\int \frac{x^2 + x + 1}{x - 1} dx & = \int (x...
... =\frac{x^2}{2} + 2x + 3\ln \vert x-1\vert + C\\
\end{aligned}\end{displaymath}


next up previous contents
下一頁: Area and Fundamental Theorem 上一頁: Integration and Its Application 前一頁: The General Power Rule   目 錄
math 2005-10-10