Generic Formulae

SSP
August 24, 2013

Quick Integrals

$\int_{-\infty }^{\infty }( odd fn) = 0 $
$\int udv = uv – \int vdu$
$\int e^{ax}dx = [\frac{1}{a}][e^{ax}]$ [a takes negative values too]
$\int xe^{ax}dx = [\frac{1}{a}][x – \frac{1}{a}][e^{ax}]$ [a takes negative values too]
$\int x^{2}e^{ax}dx = [\frac{1}{a}][(x – \frac{1}{a})^2 + \frac{1}{a^2}][e^{ax}]$ [a takes negative values too]

Quick Expansions

$\int_{-\infty }^{\infty }e^{-x^2}dx = \sqrt{pi } = \Gamma (\frac{1}{2}) $
$ \int_{0}^{\infty} x^ne^{-cx}dx = \frac{n!}{c^ { n+1}}$
$\Gamma (\alpha )= \int_{0}^{\infty}x^{\alpha-1}e^{-x}dx = (\alpha -1)!$ $ \alpha > 0$
$\sum_{0}^{\infty}\frac{x^n}{n!} = 1 + x + \frac{x^2}{2!}+ \frac{x^3}{3!}+ …\infty= e^x $

Quick Differentiation

$D(a^x) = a^x lna$ [btw]$D(e^x) = e^x$ [/btw]
$D(x^n) = nx^{n-1}$
$D(F(x)^n) = n(F(x))^{n-1}D(F(x))$
$D(ln(F(x))) = \frac{1}{F(x)}D(F(x))$