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
Referencia: Leon W Couch Apéndice p656
Integrales Indefinidas
\int (a+bx)^n dx = \frac{(a+bx)^{n+1}} {b(n+1)}, 0<n \int \frac{dx}{a+bx} =\frac{1}{b} ln|a+bx| \int \frac{dx}{(a+bx)^n} = \frac{-1}{(n-1)b(a+bx)^{n-1}} , 1<n\int \frac{dx}{(c+bc+ax^2)^n} = = \begin{cases} \frac{2}{ \sqrt{4ac-b^2}} tan^{-1}\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right) , & b^{2} < 4ac \\ \frac{1}{\sqrt{b^2-4ac}}ln\left| \frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}} \right| , & b^{2} > 4ac \\ \frac{-2}{\sqrt{2ax+b}} , & b^{2}=4ac \end{cases}
\int \frac{x dx}{c+bx+ax^2} = = \frac{1}{2a} ln\left| ax^2+bx+c \right| - \frac{b}{2a}\int \frac{dx}{c+bx+ax^2}
\int \frac{dx}{a^2+b^2x^2} = \frac{1}{ab} tan^{-1}\left( \frac{bx}{a} \right) \int \frac{x dx}{a^2+x^2} = \frac{1}{2} ln( a^2+x^2 )
Trigonométricas
\int cos(x) dx = sen(x) \int sen(x) dx = -cos(x) \int x cos(x) dx = cos(x) + x sen(x) \int x sen(x) dx = sen(x) - x cos(x) \int x^2 cos(x) dx = 2x cos(x) + (x^2 -2) sen(x) \int x^2 sen(x) dx = 2x sen(x) - (x^2 -2) cos(x)Exponenciales
\int e^{ax} dx = \frac{e^{ax}}{a} \int x e^{ax} dx = e^{ax} \left( \frac{x}{a} - \frac{1}{a^2} \right) \int x^2 e^{ax} dx = e^{ax} \left( \frac{x^2}{a} - \frac{2x}{a^2} + \frac{2}{a^3} \right) \int x^3 e^{ax} dx = e^{ax} \left( \frac{x^3}{a} - \frac{3x^2}{a^2} + \frac{6x}{a^3} - \frac{6}{a^4}\right) \int e^{ax} sen(x) dx = \frac{e^{ax}}{a^2 +1} (a sen(x) - cos(x)) \int e^{ax} cos(x) dx = \frac{e^{ax}}{a^2 +1} (a cos(x) - sen(x))