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[1]
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}
[2]
∫
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 !!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))))
[3]
∫
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}
