Definite Integrals
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| 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 \atop 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 ), \quad \text{where } c \text{ is a number between } a \text{ and } b \text{ as long as } f(x) \text{ is continous between } a \text{ and } b. $ | |
| $ \int_{a}^{b} f ( x ) g ( x ) d x = f ( c ) \int\limits_{a}^{b} g ( x ) d x, $ | |
| $ \text{where } c \text{ is a number between } a \text{ and } b \text{ as long as } f(x) \text{ is continous between } a \text{ and } b, \text{ 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} $ | |
| Definite Integral containing rational and irrational expressions | |
| $ \int_{a}^{\infty} \frac {d x}{x^2 + a^2} = \frac{\pi}{2a} $ | |
| $ \int_{0}^{\infty} \frac{x^{p-1} d x}{1 + x} = \frac{\pi}{\sin p \pi} \qquad 0<p<1 $ | |
| $ \int_{0}^{\infty} \frac{x^m d x}{x^n + a^n} = \frac{\pi a^{m+1-n}}{n \sin [ \frac{( m + 1 ) \pi}{n} ] } \qquad 0<m+1<n $ | |
| $ \int_{0}^{\infty} \frac{x^m d x}{1 + 2 x \cos \beta + x^2} = \frac{\pi}{\sin m \pi} \frac{\sin m \beta}{\sin \beta} $ | |
| $ \int_{0}^{a} \frac{dx}{\sqrt{a^2 - x^2}} = \frac {\pi}{2} $ | |
| $ \int_{0}^{a} \sqrt{a^2 - x^2} d x = \frac{\pi a^2}{4} $ | |
| $ \int_{0}^{a} x^m ( a^n - x^n )^p d x = \frac{a^{m+1+np} \Gamma [ \frac{m+1}{n} ] \Gamma ( p + 1 )}{n \Gamma [ \frac{m+1}{n} + p + 1 ]} $ | |
| $ \int_{0}^{a} \frac{x^m d x}{( a^n + x^n )^r} = \frac{ (-1)^{r-1} \pi a^{m+1-nr} \Gamma [ \frac{m+1}{n} ] }{n \sin [ \frac{(m+1)\pi}{n} ] ( r - 1 ) ! \Gamma [ \frac{m+1}{n} - r + 1]} \qquad 0<m+1<nr $ | |
| Definite Integral containing circular functions | |
| $ \int_{0}^{\pi} \sin mx \sin nx dx = \begin{cases} 0, & m=n \\ \frac{\pi}{2}, & m \neq n \end{cases}. $ | |
| $ \int_{0}^{\pi} \cos mx \cos nx dx = \begin{cases} 0, & m=n \\ \frac{\pi}{2}, & m \neq n \end{cases}. $ | |
| $ \int_{0}^{\pi} \sin mx \cos nx dx = \begin{cases} 0, & \text{if m+n is an odd number}\\ \frac{2m}{m^2-n^2}, & \text{if m+n is an even number} \end{cases} . $ | |
| $ \int_{0}^{\frac{\pi}{2}} \sin^2 x d x = \int_{0}^{\frac{\pi}{2}} \cos^2 x d x = \frac{\pi}{4} $ | |
