Referencia: Leon W Couch Apéndice p657, 658
Integrales Definidas
Definición
\int f(x) dx = \lim_{\Delta \rightarrow 0} \left( \sum_{n} \left[ f(n \Delta x)\right] \Delta x \right)Cambio de variable. Sea v=u(x)
\int_{a}^{b} f(x) dx = \int_{u(a)}^{u(b)} \left( \left. \frac{f(x)}{dv/dx} \right|_{x=u^{-1}(v)}\right) dvintegración por partes
\int u dv = uv - \int v duIntegrales Definidas
\int_{0}^{\infty} \frac{x^{m-1}}{1+x^n} dx = \frac{\pi /n}{sen(m\pi/n)}, \text{ }n>m>0\int_{0}^{\infty} x^{\alpha-1}e^{-x} dx = \Gamma(\alpha) , \alpha > 0 \text{donde: }\Gamma(\alpha +1) = \alpha \Gamma(\alpha), \Gamma (1) = 1, \Gamma [1/2] = \sqrt{\pi}, \Gamma(n) = (n-1)! \text{, si n es entero positivo }
\int_{0}^{\infty} x^{2n} e^{-ax^2} dx =\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2^{n+1}a^{n}} \sqrt{\frac{\pi}{a}} \int_{-\infty}^{\infty} e^{-a^2 x^2 + bx} dx =\frac{\sqrt{\pi}}{a} e^{b^2/(4a^2)}, a>0 \int_{0}^{\infty} e^{-ax}cos(bx) dx = \frac{a}{a^2+b^2}, a>0 \int_{0}^{\infty} e^{-ax}sen(bx) dx = \frac{b}{a^2+b^2}, a>0 \int_{0}^{\infty} e^{-a^2x^2}cos(bx) dx = \frac{\sqrt{\pi} e^{-b^2/4a^2}}{2a}, a>0
\int_{0}^{\infty} x^{\alpha-1}cos(bx) dx = \frac{\Gamma(\alpha)}{b^{\alpha}} cos \left(\frac{1}{2}\pi \alpha \right), 0<\alpha < 1, b >0
\int_{0}^{\infty} x^{\alpha-1}sen(bx) dx = \frac{\Gamma(\alpha)}{b^{\alpha}} sen \left(\frac{1}{2}\pi \alpha \right), 0<|\alpha| < 1, b >0
\int_{0}^{\infty} x e^{-ax^2} I_k(bx) dx = \frac{1}{2a} e^{b^2/4a}, \text{donde: } I_k(bx)=\frac{1}{\pi}\int_{0}^{\pi} e^{bx cos(\theta)} cos(k\theta) d\theta
\int_{0}^{\infty} \frac{sen(x)}{x} dx = \int_{0}^{\infty} Sa(x) dx = \frac{\pi}{2} \int_{0}^{\infty} \left( \frac{sen(x)}{x} \right)^2 dx = \int_{0}^{\infty} Sa^2(x) dx = \frac{\pi}{2} \int_{-\infty}^{\infty} e^{\pm j2 \pi yx} dx = \delta (y) \int_{0}^{\infty}\frac{cos(ax)}{b^2 + x^2}dx = \frac{\pi}{2b} e^{-ab}, a>0,b>0 \int_{0}^{\infty}\frac{x sen(ax)}{b^2 + x^2}dx = \frac{\pi}{2} e^{-ab}, a>0,b>0