$$x_k = a + hk$$

$$h=\frac{b-a}{N}$$

$x_0=a$、$x_N=b$

$$ f(x)=f(x_k)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)(x-x_k)^n $$

\begin{align} \int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x&=f(x_k)\left({x_{k}}-{x_{k+1}}\right)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)\int_{x_{k}}^{x_{k+1}}(x-x_k)^n\mathrm{d}x\\ &=f(x_k)h+\sum_{n=1}^{\infty}\frac{1}{(n+1)!}f^{(n)}(x_k)h^{n+1} \end{align}

\begin{align} I(f)\equiv \int_a^b f(x)\mathrm{d}x&=\sum_{k=0}^{N-1}\int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x\\ &=h\sum_{k=0}^{N-1}f(x_k)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}h\sum_{k=0}^{N-1}f^{(n)}(x_k) \end{align}

$$ R_h(f)\equiv h\sum_{k=0}^{N-1}f(x_k) $$

$$ I(f)=R_h(f)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}R_h(f^{(n)}) $$

$$ f(x)=f(x_{k+1})+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_{k+1})(x-x_{k+1})^n $$

$$ \int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x=f(x_{k+1})h-\sum_{n=1}^{\infty}\frac{1}{(n+1)!}f^{(n)}(x_{k+1})(-h)^{n+1} $$

\begin{align} I(f)&=h\sum_{k=0}^{N-1}f(x_{k+1})+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}h\sum_{k=0}^{N-1}f^{(n)}(x_{k+1})\\ &=h\sum_{k=1}^{N}f(x_{k})+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}h\sum_{k=1}^{N}f^{(n)}(x_{k})\\ &=h\sum_{k=0}^{N-1}f(x_{k})-hf(x_0)+hf(x_N)+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}\left\{h\sum_{k=0}^{N-1}f^{(n)}(x_{k})-hf^{(n)}(x_0)+hf^{(n)}(x_N)\right\} \end{align}

\begin{align} I(f)&=R_h(f)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}R_h(f^{(n)})\\ I(f)&=R_h(f)+h\left(f(b)-f(a)\right)+\sum_{n=1}^{\infty}\frac{(-h)^n}{(n+1)!}\left\{R_h(f^{(n)})+h\left(f^{(n)}(b)-f^{(n)}(a)\right)\right\}\\ \end{align}

\begin{align} I(f)&=R_h(f)+\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}R_h(f^{(2m-1)})+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}R_h(f^{(2m)})\\ I(f)&=R_h(f)+h\left(f(b)-f(a)\right) -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\left\{R_h(f^{(2m-1)})+h\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\right\}\\ &+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}\left\{R_h(f^{(2m)})+h\left(f^{(2m)}(b)-f^{(2m)}(a)\right)\right\}\\ \end{align}

\begin{align} I(f)&=R_h(f)+\frac{h}{2}\left(f(b)-f(a)\right) -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\left\{\frac{h}{2}\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\right\}\\ &+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}\left\{R_h(f^{(2m)})+\frac{h}{2}\left(f^{(2m)}(b)-f^{(2m)}(a)\right)\right\}\\ \end{align}

\begin{align} T_h(f)&\equiv R_h(f)+\frac{h}{2}\left(f(b)-f(a)\right)\\ I(f^{(n)})&=f^{(n-1)}(b)-f^{(n-1)}(a) \end{align}