# 高斯求积

${\displaystyle \int _{-1}^{1}f(x)dx=\int _{-1}^{1}W(x)g(x)dx\approx \sum _{i=1}^{n}w_{i}'g(x_{i})}$

${\displaystyle W(x)=(1-x^{2})^{-1/2}}$（高斯切比雪夫）

${\displaystyle W(x)=e^{-x^{2}}}$（高斯埃米特）。

## 高斯勒让得求积

${\displaystyle w_{i}={\frac {2}{(1-x_{i}^{2})[P_{n}'(x_{i})^{2}]}}}$

${\displaystyle x_{i}}$${\displaystyle P_{n}(x)}$的第${\displaystyle i}$個根。

${\displaystyle P_{n}(x)=\prod _{1\leq i\leq n}(x-x_{i})}$

1 0 2
2 ${\displaystyle \pm 1/{\sqrt {3}}}$ 1
3 0 89
${\displaystyle \pm {\sqrt {3/5}}}$ 59
4 ${\displaystyle \pm {\tfrac {\sqrt {525-70{\sqrt {30}}}}{35}}}$ ${\displaystyle {\tfrac {18+{\sqrt {30}}}{36}}}$
${\displaystyle \pm {\tfrac {\sqrt {525+70{\sqrt {30}}}}{35}}}$ ${\displaystyle {\tfrac {18-{\sqrt {30}}}{36}}}$
5 0 128225
${\displaystyle \pm {\tfrac {\sqrt {245-14{\sqrt {70}}}}{21}}}$ ${\displaystyle {\tfrac {322+13{\sqrt {70}}}{900}}}$
${\displaystyle \pm {\tfrac {\sqrt {245+14{\sqrt {70}}}}{21}}}$ ${\displaystyle {\tfrac {322-13{\sqrt {70}}}{900}}}$

## 变区间法则

${\displaystyle \int _{a}^{b}f(x)\,dx={\frac {b-a}{2}}\int _{-1}^{1}f\left({\frac {b-a}{2}}x+{\frac {a+b}{2}}\right)\,dx}$

${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{2}}\sum _{i=1}^{n}w_{i}f\left({\frac {b-a}{2}}x_{i}+{\frac {a+b}{2}}\right)}$

## 其他形式

${\displaystyle \int _{a}^{b}w(x)f(x)dx}$

${\displaystyle a=-1}$${\displaystyle b=1}$${\displaystyle w(x)=1}$时，即为上述内容。我们还可以用别的积分规则，如下表所示。

[−1, 1] ${\displaystyle 1\,}$ 勒让德多项式
(−1, 1) ${\displaystyle (1-x)^{\alpha }(1+x)^{\beta },\quad \alpha ,\beta >-1\,}$ 雅可比多项式
(−1, 1) ${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$ 切比雪夫多项式 (第一类)
[−1, 1] ${\displaystyle {\sqrt {1-x^{2}}}}$ 切比雪夫多项式 (第二类)
[0, ∞) ${\displaystyle e^{-x}\,}$ 拉盖尔多项式
(−∞, ∞) ${\displaystyle e^{-x^{2}}}$ 埃尔米特多项式