En estas expresiones,
![{\ Displaystyle \ phi (x) = {\ frac {1} {\ sqrt {2 \ pi}}} e ^ {- {\ frac {1} {2}} x ^ {2}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
es la función de densidad de probabilidad normal estándar ,
![{\ Displaystyle \ Phi (x) = \ int _ {- \ infty} ^ {x} \ phi (t) \, dt = {\ frac {1} {2}} \ left (1+ \ operatorname {erf} \ left ({\ frac {x} {\ sqrt {2}}} \ right) \ right)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
es la función de distribución acumulativa correspondiente (donde erf es la función de error ) y
![T (h, a) = \ phi (h) \ int _ {0} ^ {a} {\ frac {\ phi (hx)} {1 + x ^ {2}}} \, dx](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
es la función T de Owen .
Owen [nb 1] tiene una extensa lista de integrales de tipo gaussiano; a continuación se proporciona solo un subconjunto.
![\int \phi (x)\,dx=\Phi (x)+C](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![\int x\phi (x)\,dx=-\phi (x)+C](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![\int x^{2}\phi (x)\,dx=\Phi (x)-x\phi (x)+C](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
[nb 2]![\int x^{{2k+2}}\phi (x)\,dx=-\phi (x)\sum _{{j=0}}^{k}{\frac {(2k+1)!!}{(2j+1)!!}}x^{{2j+1}}+(2k+1)!!\,\Phi (x)+C](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
En estas integrales, n !! es el factorial doble : para n pares es igual al producto de todos los números pares de 2 an , y para n impares es el producto de todos los números impares de 1 an ; adicionalmente se asume que 0 !! = (−1) !! = 1 .
![{\displaystyle \int \phi (x)^{2}\,dx={\frac {1}{2{\sqrt {\pi }}}}\Phi \left(x{\sqrt {2}}\right)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
[nb 3]![{\displaystyle \int x\phi (a+bx)\,dx=-{\frac {1}{b^{2}}}\left(\phi (a+bx)+a\Phi (a+bx)\right)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{2}\phi (a+bx)\,dx={\frac {1}{b^{3}}}\left((a^{2}+1)\Phi (a+bx)+(a-bx)\phi (a+bx)\right)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![\int \phi (a+bx)^{n}\,dx={\frac {1}{b{\sqrt {n(2\pi )^{{n-1}}}}}}\Phi \left({\sqrt {n}}(a+bx)\right)+C](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \Phi (a+bx)\,dx={\frac {1}{b}}\left((a+bx)\Phi (a+bx)+\phi (a+bx)\right)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x\Phi (a+bx)\,dx={\frac {1}{2b^{2}}}\left((b^{2}x^{2}-a^{2}-1)\Phi (a+bx)+(bx-a)\phi (a+bx)\right)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{2}\Phi (a+bx)\,dx={\frac {1}{3b^{3}}}\left((b^{3}x^{3}+a^{3}+3a)\Phi (a+bx)+(b^{2}x^{2}-abx+a^{2}+2)\phi (a+bx)\right)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![\int x^{n}\Phi (x)\,dx={\frac {1}{n+1}}\left(\left(x^{{n+1}}-nx^{{n-1}}\right)\Phi (x)+x^{n}\phi (x)+n(n-1)\int x^{{n-2}}\Phi (x)\,dx\right)+C](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x\phi (x)\Phi (a+bx)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(xt+{\frac {ab}{t}}\right)-\phi (x)\Phi (a+bx)+C,\qquad t={\sqrt {1+b^{2}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \Phi (x)^{2}\,dx=x\Phi (x)^{2}+2\Phi (x)\phi (x)-{\frac {1}{\sqrt {\pi }}}\Phi \left(x{\sqrt {2}}\right)+C}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![\int e^{{cx}}\phi (bx)^{n}\,dx={\frac {e^{{{\frac {c^{2}}{2nb^{2}}}}}}{b{\sqrt {n(2\pi )^{{n-1}}}}}}\Phi \left({\frac {b^{2}xn-c}{b{\sqrt {n}}}}\right)+C,\qquad b\neq 0,n>0](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)