| Table of Definite Integrals | |
|---|---|
| Definition of Definite Integral | |
| $ \int\limits_{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\limits_{a}^{b} f ( x ) d x = \int\limits_{a}^{b} \frac{d}{dx} g ( x ) d x = g ( x ) |_{a}^{b} = g ( b ) - g ( a ) $ | |
| $ \int\limits_{a}^{\infty} d x = \lim_{n \to \infty} \int\limits_{a}^{b} f ( x ) d x $ | |
| $ \int\limits_{-\infty}^{\infty} f ( x ) d x = \lim_{a \to - \infty \ b \to \infty} \int\limits_{a}^{b} f ( x ) d x $ | |
| $ \int\limits_{a}^{b} f ( x ) d x = \lim_{\epsilon \to \infty} \int\limits_{a}^{b - \epsilon} f ( x ) d x $ | |
| $ \int\limits_{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\limits_{a}^{b} c f ( x ) d x = c \int\limits_{a}^{b} f ( x ) d x $ | |
| $ \int\limits_{a}^{a} f ( x ) d x = 0 $ | |
| $ \int\limits_{a}^{b} f ( x ) d x = - \int\limits{b}^{a} f ( x ) d x $ | |
| $ \int\limits_{a}^{b} f ( x ) d x = \int\limits_{a}^{c} f ( x ) d x + \int\limits_{c}^{b} f ( x ) d x $ | |
| $ \int\limits_{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\limits_{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 $ | |
