Formelsammlung Mathematik: Numerische Integration

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx h\sum _{k=0}^{n-1}y_{k}}$

mit ${\displaystyle h={\frac {b-1}{n}}}$ und ${\displaystyle y_{k}=f(a+kh)}$.

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx (b-a)f\left({\frac {a+b}{2}}\right).}$

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx (b-a){\frac {f(a)+f(b)}{2}}.}$

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx {\frac {b-a}{6}}\left[f(a)+f\left({\frac {a+b}{2}}\right)+f(b)\right].}$

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx {\frac {3h}{8}}(y_{0}+3y_{1}+3y_{2}+y_{3})}$

mit ${\displaystyle h={\frac {b-a}{3}}}$ und ${\displaystyle y_{k}=f(a+kh)}$.

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx {\frac {b-a}{90}}(7y_{0}+32y_{1}+12y_{2}+32y_{3}+7y_{4})}$

mit ${\displaystyle h={\frac {b-a}{4}}}$ und ${\displaystyle y_{k}=f(a+kh)}$.

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx {\frac {h}{3}}{\bigg [}y_{0}+4{\bigg (}\sum _{k=1}^{n/2}y_{2k-1}{\bigg )}+2{\bigg (}\sum _{k=1}^{n/2-1}y_{2k}{\bigg )}+y_{n}{\bigg ]}}$

mit ${\displaystyle h={\frac {b-a}{n}}}$ und ${\displaystyle y_{k}=f(a+kh)}$.

${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x\approx {\frac {b-a}{840}}(41y_{0}+216y_{1}+27y_{2}+272y_{3}+27y_{4}+216y_{5}+41y_{6})}$

mit ${\displaystyle h={\frac {b-a}{6}}}$ und ${\displaystyle y_{k}=f(a+kh)}$.