在數值分析 上，梯形法則 和辛卜生法則 均是數值積分 的方法。它們都是計算定積分 的。
這兩種方法都屬於牛頓-寇次公式 。它們以函數於等距
n
+
1
{\displaystyle n+1}
n
{\displaystyle n}
多項式 來近似原來的函數，再行求積。
原函數（藍色）近似為紅色的線性函數 多重梯形法則 梯形法則 是：
∫
a
b
f
(
x
)
d
x
≈
(
b
−
a
)
f
(
a
)
+
f
(
b
)
2
.
{\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a){\frac {f(a)+f(b)}{2}}.}
這等同將被積函數近似為直線 函數，被積的部分近似為梯形 。
要求得較準確的數值，可以將要求積的區間分成多個小區間，再個別估計，即：
∫
a
b
f
(
x
)
d
x
≈
b
−
a
n
(
f
(
a
)
+
f
(
b
)
2
+
∑
k
=
1
n
−
1
f
(
a
+
k
b
−
a
n
)
)
.
{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{n}}\left({f(a)+f(b) \over 2}+\sum _{k=1}^{n-1}f\left(a+k{\frac {b-a}{n}}\right)\right).}
可改寫成
∫
a
b
f
(
x
)
d
x
≈
b
−
a
2
n
(
f
(
x
0
)
+
2
f
(
x
1
)
+
2
f
(
x
2
)
+
⋯
+
2
f
(
x
n
−
1
)
+
f
(
x
n
)
)
{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{2n}}\left(f(x_{0})+2f(x_{1})+2f(x_{2})+\cdots +2f(x_{n-1})+f(x_{n})\right)}
其中
對
k
=
0
,
1
,
…
,
n
{\displaystyle k=0,1,\dots ,n}
x
k
=
a
+
k
b
−
a
n
,
{\displaystyle x_{k}=a+k{\frac {b-a}{n}},}
辛卜生法則 （Simpson's rule，又稱森遜法則 ）是：
∫
a
b
f
(
x
)
d
x
≈
b
−
a
6
[
f
(
a
)
+
4
f
(
a
+
b
2
)
+
f
(
b
)
]
.
{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right].}
同樣地，辛卜生法則也有多重的版本：
∫
a
b
f
(
x
)
d
x
≈
h
3
⋅
[
f
(
x
0
)
+
2
∑
k
=
1
n
−
1
f
(
x
k
)
+
4
∑
k
=
1
n
f
(
x
k
−
1
+
x
k
2
)
+
f
(
x
n
)
]
{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}\cdot \left[f(x_{0})+2\sum _{k=1}^{n-1}f(x_{k})+4\sum _{k=1}^{n}f\left({\frac {x_{k-1}+x_{k}}{2}}\right)+f(x_{n})\right]}
h
=
b
−
a
n
,
x
k
=
a
+
k
⋅
h
.
{\displaystyle h={\frac {b-a}{n}},\ x_{k}=a+k\cdot h.}
或寫成
∫
a
b
f
(
x
)
d
x
≈
h
3
[
f
(
x
0
)
+
4
f
(
x
1
)
+
2
f
(
x
2
)
+
4
f
(
x
3
)
+
2
f
(
x
4
)
+
⋯
+
4
f
(
x
n
−
1
)
+
f
(
x
n
)
]
{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots +4f(x_{n-1})+f(x_{n}){\bigg ]}}
牛頓-寇次公式（Newton-Cotes rule / Newton-Cotes formula）以Roger Cotes和艾薩克·牛頓 命名。其內容是：
∫
a
b
f
(
x
)
d
x
≈
∑
i
=
0
n
w
i
f
(
x
i
)
{\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{i=0}^{n}w_{i}\,f(x_{i})}
其中對
k
=
0
,
1
,
…
,
n
{\displaystyle k=0,1,\dots ,n}
w
i
{\displaystyle w_{i}}
n
{\displaystyle n}
x
k
=
a
+
k
b
−
a
n
{\displaystyle x_{k}=a+k{\frac {b-a}{n}}}
梯形法則和辛卜生法則便是
n
=
1
,
2
{\displaystyle n=1,2}
亦有不採用在邊界點來估計的版本，即取
x
k
=
a
+
k
b
−
a
n
+
1
{\displaystyle x_{k}=a+k{\frac {b-a}{n+1}}}
假設已知
f
(
x
0
)
,
f
(
x
1
)
,
…
,
f
(
x
n
)
{\displaystyle f(x_{0}),f(x_{1}),\dots ,f(x_{n})}
以
n
+
1
{\displaystyle n+1}
插值 ，求得對應
f
(
x
)
{\displaystyle f(x)}
拉格朗日多項式 。
對該
n
{\displaystyle n}
該積分便可以作為
∫
a
b
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)\,dx}
n
{\displaystyle n}
w
i
{\displaystyle w_{i}}
對於次數較高的多項式而有很大誤差（龍格現象 ），不如高斯積分法 。
下表中
f
i
=
f
(
x
i
)
{\displaystyle f_{i}=f(x_{i})}
ξ
∈
[
a
,
b
]
{\displaystyle \xi \in [a,b]}
h
=
b
−
a
{\displaystyle h=b-a}
精度
名稱
公式
誤差
1
梯形法則
h
2
(
f
0
+
f
1
)
{\displaystyle {\frac {h}{2}}(f_{0}+f_{1})}
−
2
h
3
3
f
(
2
)
(
ξ
)
{\displaystyle -{\frac {2h^{3}}{3}}\,f^{(2)}(\xi )}
2
辛卜生法則
h
6
(
f
0
+
4
f
1
+
f
2
)
{\displaystyle {\frac {h}{6}}(f_{0}+4f_{1}+f_{2})}
−
h
5
90
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {h^{5}}{90}}\,f^{(4)}(\xi )}
3
辛卜生3/8法則
h
8
(
f
0
+
3
f
1
+
3
f
2
+
f
3
)
{\displaystyle {\frac {h}{8}}(f_{0}+3f_{1}+3f_{2}+f_{3})}
−
3
h
5
80
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {3h^{5}}{80}}\,f^{(4)}(\xi )}
4
保爾法則
2
h
45
(
7
f
0
+
32
f
1
+
12
f
2
+
32
f
3
+
7
f
4
)
{\displaystyle {\frac {2h}{45}}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})}
−
8
h
7
945
f
(
6
)
(
ξ
)
{\displaystyle -{\frac {8h^{7}}{945}}\,f^{(6)}(\xi )}
不用界點的
0
中點法
2
h
f
1
{\displaystyle 2hf_{1}\,}
h
3
24
f
(
2
)
(
ξ
)
{\displaystyle {\frac {h^{3}}{24}}\,f^{(2)}(\xi )}
1
3
h
2
(
f
1
+
f
2
)
{\displaystyle {\frac {3h}{2}}(f_{1}+f_{2})}
h
3
4
f
(
2
)
(
ξ
)
{\displaystyle {\frac {h^{3}}{4}}\,f^{(2)}(\xi )}
2
4
h
3
(
2
f
1
−
f
2
+
2
f
3
)
{\displaystyle {\frac {4h}{3}}(2f_{1}-f_{2}+2f_{3})}
28
h
5
90
f
(
4
)
(
ξ
)
{\displaystyle {\frac {28h^{5}}{90}}f^{(4)}(\xi )}
3
5
h
24
(
11
f
1
+
f
2
+
f
3
+
11
f
4
)
{\displaystyle {\frac {5h}{24}}(11f_{1}+f_{2}+f_{3}+11f_{4})}
95
h
5
144
f
(
4
)
(
ξ
)
{\displaystyle {\frac {95h^{5}}{144}}f^{(4)}(\xi )}
M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . New York: Dover, 1972. (See Section 25.4.)
George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations . Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Section 5.1.)
William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C . Cambridge, UK: Cambridge University Press, 1988. (See Section 4.1.)
Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis . New York: Springer-Verlag, 1980. (See Section 3.1.)
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c803efa545e470e97be928947d5d97c06e46553a)
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}\cdot \left[f(x_{0})+2\sum _{k=1}^{n-1}f(x_{k})+4\sum _{k=1}^{n}f\left({\frac {x_{k-1}+x_{k}}{2}}\right)+f(x_{n})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1453643a132e141dc53b82b2b77479d386c288)
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots +4f(x_{n-1})+f(x_{n}){\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8029db8932a90d0c6489e71bb594924cb546065e)
![{\displaystyle \xi \in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
