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! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Definition of Definite Integral | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Definition of Definite Integral | ||
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| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) d x = \lim_{n \to \infty} { f ( a ) \Delta x + f ( a + \Delta x ) \Delta x + f ( a + 2 \Delta x ) + \cdot \cdot \cdot + f ( a + ( n - 1 ) \Delta x ) \Delta x }</math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) d x = \int_{a}^{b} \frac{d}{dx} g ( x ) d x = g ( x ) |_{a}^{b} = g ( b ) - g ( a ) </math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{\infty} d x = \lim_{n \to \infty} \int\limits_{a}^{b} f ( x ) d x </math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{-\infty}^{\infty} f ( x ) d x = \lim_{a \to - \infty \ b \to \infty} \int\limits_{a}^{b} f ( x ) d x</math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) d x = \lim_{\epsilon \to \infty} \int\limits_{a}^{b - \epsilon} f ( x ) d x</math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) d x = \lim_{\epsilon \to \infty} \int\limits_{a + \epsilon}^{b} f ( x ) d x</math> |
|- | |- | ||
! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | General Rules for Definite Integral | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | General Rules for Definite Integral | ||
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|<math> \int\limits_{a}^{b} { f ( x ) \pm g ( x ) \pm h ( x ) \pm \cdot \cdot \cdot } d x = \int\limits_{a}^{b} f ( x ) d x \pm \int\limits_{a}^{b} g ( x ) d x \pm \int\limits_{a}^{b} h ( x ) d x \pm \cdot \cdot \cdot</math> | |<math> \int\limits_{a}^{b} { f ( x ) \pm g ( x ) \pm h ( x ) \pm \cdot \cdot \cdot } d x = \int\limits_{a}^{b} f ( x ) d x \pm \int\limits_{a}^{b} g ( x ) d x \pm \int\limits_{a}^{b} h ( x ) d x \pm \cdot \cdot \cdot</math> | ||
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} c f ( x ) d x = c \int_{a}^{b} f ( x ) d x </math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{a} f ( x ) d x = 0 </math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) d x = - \int_{b}^{a} f ( x ) d x </math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) d x = \int_{a}^{c} f ( x ) d x + \int_{c}^{b} f ( x ) d x </math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) d x = ( b - a ) f ( c ) \qquad c \ is \ a \ number \ between \ a \ and \ b \ as \ long \ as \ f ( x ) \ is \ continous \ between \ a \ and \ b </math> |
|- | |- | ||
| − | |<math> \ | + | |<math> \int_{a}^{b} f ( x ) g ( x ) d x = f ( c ) \int\limits_{a}^{b} g ( x ) d x \qquad c \ is \ a \ number \ between \ a \ and \ b \ as \ long \ as \ f ( x ) \ is \ continous \ between \ a \ and \ b </math> |
| + | |- | ||
| + | |<math> \ and \ g ( x ) \ge 0 </math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Leibnitz rule for derivation | ||
| + | |- | ||
| + | |<math> \frac{d}{d \alpha} \int_{\Phi_1 ( \alpha )}^{\Phi_2 ( \alpha ) } F ( x , \alpha ) d x = \int_{\Phi_1 ( \alpha )}^{\Phi_2 ( \alpha ) } \frac{\partial F}{\partial \alpha} d x + F ( \Phi_2 , \alpha ) \frac{d \Phi_1}{d \alpha} - F ( \Phi_1 , \alpha ) \frac{d \Phi_2}{d \alpha}</math> | ||
Revision as of 22:14, 17 November 2010
| Table of Definite Integrals | |
|---|---|
| Definition of Definite Integral | |
| $ \int_{a}^{b} f ( x ) d x = \lim_{n \to \infty} { f ( a ) \Delta x + f ( a + \Delta x ) \Delta x + f ( a + 2 \Delta x ) + \cdot \cdot \cdot + f ( a + ( n - 1 ) \Delta x ) \Delta x } $ | |
| $ \int_{a}^{b} f ( x ) d x = \int_{a}^{b} \frac{d}{dx} g ( x ) d x = g ( x ) |_{a}^{b} = g ( b ) - g ( a ) $ | |
| $ \int_{a}^{\infty} d x = \lim_{n \to \infty} \int\limits_{a}^{b} f ( x ) d x $ | |
| $ \int_{-\infty}^{\infty} f ( x ) d x = \lim_{a \to - \infty \ b \to \infty} \int\limits_{a}^{b} f ( x ) d x $ | |
| $ \int_{a}^{b} f ( x ) d x = \lim_{\epsilon \to \infty} \int\limits_{a}^{b - \epsilon} f ( x ) d x $ | |
| $ \int_{a}^{b} f ( x ) d x = \lim_{\epsilon \to \infty} \int\limits_{a + \epsilon}^{b} f ( x ) d x $ | |
| General Rules for Definite Integral | |
| $ \int\limits_{a}^{b} { f ( x ) \pm g ( x ) \pm h ( x ) \pm \cdot \cdot \cdot } d x = \int\limits_{a}^{b} f ( x ) d x \pm \int\limits_{a}^{b} g ( x ) d x \pm \int\limits_{a}^{b} h ( x ) d x \pm \cdot \cdot \cdot $ | |
| $ \int_{a}^{b} c f ( x ) d x = c \int_{a}^{b} f ( x ) d x $ | |
| $ \int_{a}^{a} f ( x ) d x = 0 $ | |
| $ \int_{a}^{b} f ( x ) d x = - \int_{b}^{a} f ( x ) d x $ | |
| $ \int_{a}^{b} f ( x ) d x = \int_{a}^{c} f ( x ) d x + \int_{c}^{b} f ( x ) d x $ | |
| $ \int_{a}^{b} f ( x ) d x = ( b - a ) f ( c ) \qquad c \ is \ a \ number \ between \ a \ and \ b \ as \ long \ as \ f ( x ) \ is \ continous \ between \ a \ and \ b $ | |
| $ \int_{a}^{b} f ( x ) g ( x ) d x = f ( c ) \int\limits_{a}^{b} g ( x ) d x \qquad c \ is \ a \ number \ between \ a \ and \ b \ as \ long \ as \ f ( x ) \ is \ continous \ between \ a \ and \ b $ | |
| $ \ and \ g ( x ) \ge 0 $ | |
| Leibnitz rule for derivation | |
| $ \frac{d}{d \alpha} \int_{\Phi_1 ( \alpha )}^{\Phi_2 ( \alpha ) } F ( x , \alpha ) d x = \int_{\Phi_1 ( \alpha )}^{\Phi_2 ( \alpha ) } \frac{\partial F}{\partial \alpha} d x + F ( \Phi_2 , \alpha ) \frac{d \Phi_1}{d \alpha} - F ( \Phi_1 , \alpha ) \frac{d \Phi_2}{d \alpha} $ | |
