In the expressions in this article,
φ
(
x
)
=
1
2
π
e
−
1
2
x
2
{\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}}
is the standard normal probability density function,
Φ
(
x
)
=
∫
−
∞
x
φ
(
t
)
d
t
=
1
2
(
1
+
erf
(
x
2
)
)
{\displaystyle \Phi (x)=\int _{-\infty }^{x}\varphi (t)\,dt={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)}
is the corresponding cumulative distribution function (where erf is the error function ), and
T
(
h
,
a
)
=
φ
(
h
)
∫
0
a
φ
(
h
x
)
1
+
x
2
d
x
{\displaystyle T(h,a)=\varphi (h)\int _{0}^{a}{\frac {\varphi (hx)}{1+x^{2}}}\,dx}
is Owen's T function .
Owen has an extensive list of Gaussian-type integrals; only a subset is given below.
Indefinite integrals
∫
φ
(
x
)
d
x
=
Φ
(
x
)
+
C
{\displaystyle \int \varphi (x)\,dx=\Phi (x)+C}
∫
x
φ
(
x
)
d
x
=
−
φ
(
x
)
+
C
{\displaystyle \int x\varphi (x)\,dx=-\varphi (x)+C}
∫
x
2
φ
(
x
)
d
x
=
Φ
(
x
)
−
x
φ
(
x
)
+
C
{\displaystyle \int x^{2}\varphi (x)\,dx=\Phi (x)-x\varphi (x)+C}
∫
x
2
k
+
1
φ
(
x
)
d
x
=
−
φ
(
x
)
∑
j
=
0
k
(
2
k
)
!
!
(
2
j
)
!
!
x
2
j
+
C
{\displaystyle \int x^{2k+1}\varphi (x)\,dx=-\varphi (x)\sum _{j=0}^{k}{\frac {(2k)!!}{(2j)!!}}x^{2j}+C}
∫
x
2
k
+
2
φ
(
x
)
d
x
=
−
φ
(
x
)
∑
j
=
0
k
(
2
k
+
1
)
!
!
(
2
j
+
1
)
!
!
x
2
j
+
1
+
(
2
k
+
1
)
!
!
Φ
(
x
)
+
C
{\displaystyle \int x^{2k+2}\varphi (x)\,dx=-\varphi (x)\sum _{j=0}^{k}{\frac {(2k+1)!!}{(2j+1)!!}}x^{2j+1}+(2k+1)!!\,\Phi (x)+C}
In the previous two integrals, n !! is the double factorial : for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!! = (−1)!! = 1.
∫
φ
(
x
)
2
d
x
=
1
2
π
Φ
(
x
2
)
+
C
{\displaystyle \int \varphi (x)^{2}\,dx={\frac {1}{2{\sqrt {\pi }}}}\Phi \left(x{\sqrt {2}}\right)+C}
∫
φ
(
x
)
φ
(
a
+
b
x
)
d
x
=
1
t
φ
(
a
t
)
Φ
(
t
x
+
a
b
t
)
+
C
,
t
=
1
+
b
2
{\displaystyle \int \varphi (x)\varphi (a+bx)\,dx={\frac {1}{t}}\varphi \left({\frac {a}{t}}\right)\Phi \left(tx+{\frac {ab}{t}}\right)+C,\qquad t={\sqrt {1+b^{2}}}}
∫
x
φ
(
a
+
b
x
)
d
x
=
−
1
b
2
(
φ
(
a
+
b
x
)
+
a
Φ
(
a
+
b
x
)
)
+
C
{\displaystyle \int x\varphi (a+bx)\,dx=-{\frac {1}{b^{2}}}\left(\varphi (a+bx)+a\Phi (a+bx)\right)+C}
∫
x
2
φ
(
a
+
b
x
)
d
x
=
1
b
3
(
(
a
2
+
1
)
Φ
(
a
+
b
x
)
+
(
a
−
b
x
)
φ
(
a
+
b
x
)
)
+
C
{\displaystyle \int x^{2}\varphi (a+bx)\,dx={\frac {1}{b^{3}}}\left((a^{2}+1)\Phi (a+bx)+(a-bx)\varphi (a+bx)\right)+C}
∫
φ
(
a
+
b
x
)
n
d
x
=
1
b
n
(
2
π
)
n
−
1
Φ
(
n
(
a
+
b
x
)
)
+
C
{\displaystyle \int \varphi (a+bx)^{n}\,dx={\frac {1}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\sqrt {n}}(a+bx)\right)+C}
∫
Φ
(
a
+
b
x
)
d
x
=
1
b
(
(
a
+
b
x
)
Φ
(
a
+
b
x
)
+
φ
(
a
+
b
x
)
)
+
C
{\displaystyle \int \Phi (a+bx)\,dx={\frac {1}{b}}\left((a+bx)\Phi (a+bx)+\varphi (a+bx)\right)+C}
∫
x
Φ
(
a
+
b
x
)
d
x
=
1
2
b
2
(
(
b
2
x
2
−
a
2
−
1
)
Φ
(
a
+
b
x
)
+
(
b
x
−
a
)
φ
(
a
+
b
x
)
)
+
C
{\displaystyle \int x\Phi (a+bx)\,dx={\frac {1}{2b^{2}}}\left((b^{2}x^{2}-a^{2}-1)\Phi (a+bx)+(bx-a)\varphi (a+bx)\right)+C}
∫
x
2
Φ
(
a
+
b
x
)
d
x
=
1
3
b
3
(
(
b
3
x
3
+
a
3
+
3
a
)
Φ
(
a
+
b
x
)
+
(
b
2
x
2
−
a
b
x
+
a
2
+
2
)
φ
(
a
+
b
x
)
)
+
C
{\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)\varphi (a+bx)\right)+C}
∫
x
n
Φ
(
x
)
d
x
=
1
n
+
1
(
(
x
n
+
1
−
n
x
n
−
1
)
Φ
(
x
)
+
x
n
φ
(
x
)
+
n
(
n
−
1
)
∫
x
n
−
2
Φ
(
x
)
d
x
)
+
C
{\displaystyle \int x^{n}\Phi (x)\,dx={\frac {1}{n+1}}\left(\left(x^{n+1}-nx^{n-1}\right)\Phi (x)+x^{n}\varphi (x)+n(n-1)\int x^{n-2}\Phi (x)\,dx\right)+C}
∫
x
φ
(
x
)
Φ
(
a
+
b
x
)
d
x
=
b
t
φ
(
a
t
)
Φ
(
x
t
+
a
b
t
)
−
φ
(
x
)
Φ
(
a
+
b
x
)
+
C
,
t
=
1
+
b
2
{\displaystyle \int x\varphi (x)\Phi (a+bx)\,dx={\frac {b}{t}}\varphi \left({\frac {a}{t}}\right)\Phi \left(xt+{\frac {ab}{t}}\right)-\varphi (x)\Phi (a+bx)+C,\qquad t={\sqrt {1+b^{2}}}}
∫
Φ
(
x
)
2
d
x
=
x
Φ
(
x
)
2
+
2
Φ
(
x
)
φ
(
x
)
−
1
π
Φ
(
x
2
)
+
C
{\displaystyle \int \Phi (x)^{2}\,dx=x\Phi (x)^{2}+2\Phi (x)\varphi (x)-{\frac {1}{\sqrt {\pi }}}\Phi \left(x{\sqrt {2}}\right)+C}
∫
e
c
x
φ
(
b
x
)
n
d
x
=
e
c
2
2
n
b
2
b
n
(
2
π
)
n
−
1
Φ
(
b
2
x
n
−
c
b
n
)
+
C
,
b
≠
0
,
n
>
0
{\displaystyle \int e^{cx}\varphi (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}
Definite integrals
∫
−
∞
∞
x
2
φ
(
x
)
n
d
x
=
1
n
3
(
2
π
)
n
−
1
{\displaystyle \int _{-\infty }^{\infty }x^{2}\varphi (x)^{n}\,dx={\frac {1}{\sqrt {n^{3}(2\pi )^{n-1}}}}}
∫
−
∞
∞
φ
(
x
)
φ
(
a
+
b
x
)
d
x
=
1
1
+
b
2
φ
(
a
1
+
b
2
)
{\displaystyle \int _{-\infty }^{\infty }\varphi (x)\varphi (a+bx)\,dx={\frac {1}{\sqrt {1+b^{2}}}}\varphi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)}
∫
−
∞
0
φ
(
a
x
)
Φ
(
b
x
)
d
x
=
1
2
π
|
a
|
(
π
2
−
arctan
(
b
|
a
|
)
)
{\displaystyle \int _{-\infty }^{0}\varphi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}-\arctan \left({\frac {b}{|a|}}\right)\right)}
∫
0
∞
φ
(
a
x
)
Φ
(
b
x
)
d
x
=
1
2
π
|
a
|
(
π
2
+
arctan
(
b
|
a
|
)
)
{\displaystyle \int _{0}^{\infty }\varphi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}+\arctan \left({\frac {b}{|a|}}\right)\right)}
∫
0
∞
x
φ
(
x
)
Φ
(
b
x
)
d
x
=
1
2
2
π
(
1
+
b
1
+
b
2
)
{\displaystyle \int _{0}^{\infty }x\varphi (x)\Phi (bx)\,dx={\frac {1}{2{\sqrt {2\pi }}}}\left(1+{\frac {b}{\sqrt {1+b^{2}}}}\right)}
∫
0
∞
x
2
φ
(
x
)
Φ
(
b
x
)
d
x
=
1
4
+
1
2
π
(
b
1
+
b
2
+
arctan
(
b
)
)
{\displaystyle \int _{0}^{\infty }x^{2}\varphi (x)\Phi (bx)\,dx={\frac {1}{4}}+{\frac {1}{2\pi }}\left({\frac {b}{1+b^{2}}}+\arctan(b)\right)}
∫
−
∞
∞
x
φ
(
x
)
2
Φ
(
x
)
d
x
=
1
4
π
3
{\displaystyle \int _{-\infty }^{\infty }x\varphi (x)^{2}\Phi (x)\,dx={\frac {1}{4\pi {\sqrt {3}}}}}
∫
0
∞
Φ
(
b
x
)
2
φ
(
x
)
d
x
=
1
2
π
(
arctan
(
b
)
+
arctan
1
+
2
b
2
)
{\displaystyle \int _{0}^{\infty }\Phi (bx)^{2}\varphi (x)\,dx={\frac {1}{2\pi }}\left(\arctan(b)+\arctan {\sqrt {1+2b^{2}}}\right)}
∫
−
∞
∞
Φ
(
a
+
b
x
)
2
φ
(
x
)
d
x
=
Φ
(
a
1
+
b
2
)
−
2
T
(
a
1
+
b
2
,
1
1
+
2
b
2
)
{\displaystyle \int _{-\infty }^{\infty }\Phi (a+bx)^{2}\varphi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)-2T\left({\frac {a}{\sqrt {1+b^{2}}}},{\frac {1}{\sqrt {1+2b^{2}}}}\right)}
∫
−
∞
∞
x
Φ
(
a
+
b
x
)
2
φ
(
x
)
d
x
=
2
b
1
+
b
2
φ
(
a
t
)
Φ
(
a
1
+
b
2
1
+
2
b
2
)
{\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)^{2}\varphi (x)\,dx={\frac {2b}{\sqrt {1+b^{2}}}}\varphi \left({\frac {a}{t}}\right)\Phi \left({\frac {a}{{\sqrt {1+b^{2}}}{\sqrt {1+2b^{2}}}}}\right)}
∫
−
∞
∞
Φ
(
b
x
)
2
φ
(
x
)
d
x
=
1
π
arctan
1
+
2
b
2
{\displaystyle \int _{-\infty }^{\infty }\Phi (bx)^{2}\varphi (x)\,dx={\frac {1}{\pi }}\arctan {\sqrt {1+2b^{2}}}}
∫
−
∞
∞
x
φ
(
x
)
Φ
(
b
x
)
d
x
=
∫
−
∞
∞
x
φ
(
x
)
Φ
(
b
x
)
2
d
x
=
b
2
π
(
1
+
b
2
)
{\displaystyle \int _{-\infty }^{\infty }x\varphi (x)\Phi (bx)\,dx=\int _{-\infty }^{\infty }x\varphi (x)\Phi (bx)^{2}\,dx={\frac {b}{\sqrt {2\pi (1+b^{2})}}}}
∫
−
∞
∞
Φ
(
a
+
b
x
)
φ
(
x
)
d
x
=
Φ
(
a
1
+
b
2
)
{\displaystyle \int _{-\infty }^{\infty }\Phi (a+bx)\varphi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)}
∫
−
∞
∞
x
Φ
(
a
+
b
x
)
φ
(
x
)
d
x
=
b
t
φ
(
a
t
)
,
t
=
1
+
b
2
{\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)\varphi (x)\,dx={\frac {b}{t}}\varphi \left({\frac {a}{t}}\right),\qquad t={\sqrt {1+b^{2}}}}
∫
0
∞
x
Φ
(
a
+
b
x
)
φ
(
x
)
d
x
=
b
t
φ
(
a
t
)
Φ
(
−
a
b
t
)
+
1
2
π
Φ
(
a
)
,
t
=
1
+
b
2
{\displaystyle \int _{0}^{\infty }x\Phi (a+bx)\varphi (x)\,dx={\frac {b}{t}}\varphi \left({\frac {a}{t}}\right)\Phi \left(-{\frac {ab}{t}}\right)+{\frac {1}{\sqrt {2\pi }}}\Phi (a),\qquad t={\sqrt {1+b^{2}}}}
∫
−
∞
∞
ln
(
x
2
)
1
σ
φ
(
x
σ
)
d
x
=
ln
(
σ
2
)
−
γ
−
ln
2
≈
ln
(
σ
2
)
−
1.27036
{\displaystyle \int _{-\infty }^{\infty }\ln(x^{2}){\frac {1}{\sigma }}\varphi \left({\frac {x}{\sigma }}\right)\,dx=\ln(\sigma ^{2})-\gamma -\ln 2\approx \ln(\sigma ^{2})-1.27036}
References
Owen 1980 .
Patel & Read (1996) lists this integral without the minus sign, which is an error. See calculation by WolframAlpha .
Patel & Read (1996) report this integral with error, see WolframAlpha .
Patel & Read (1996) report this integral incorrectly by omitting x from the integrand.
Owen, D. (1980). "A table of normal integrals". Communications in Statistics: Simulation and Computation . B9 (4): 389–419. doi :10.1080/03610918008812164 . Patel, Jagdish K.; Read, Campbell B. (1996). Handbook of the normal distribution (2nd ed.). CRC Press. ISBN 0-8247-9342-0 Categories :
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
**DISCLAIMER** We are not affiliated with Wikipedia, and Cloudflare.
The information presented on this site is for general informational purposes only and does not constitute medical advice.
You should always have a personal consultation with a healthcare professional before making changes to your diet, medication, or exercise routine.
AI helps with the correspondence in our chat.
We participate in an affiliate program. If you buy something through a link, we may earn a commission 💕
↑
