Let$P\left(x\right)$ be a polynomial of degree $n$ with leading 

coefficient 1 . Let $v\left(x\right)$ be any function and $v_1\left(x\right)=\int v\left(x\right)\mathrm{d}x$

$v_2\left(x\right)=\int v_1\left(x\right)\mathrm{d}x\dots v_{n+1}\left(x\right)=\int v_n\left(x\right)\mathrm{d}x$ then $\int P\left(x\right)v\left(x\right)dx$ is equal to 

$P\left(x\right)v_1\left(x\right)+\frac{P^{\mathrm{\prime }}\left(x\right)v_2\left(x\right)}{2!}+\frac{P^{\prime \prime }\left(x\right)v_3\left(x\right)}{3!}\dots +\frac{v_{n+1}\left(x\right)}{(n+1)!}$

$P\left(x\right)v_1\left(x\right)-P^{\mathrm{\prime }}\left(x\right)v_2\left(x\right)+P^{\prime \prime }\left(x\right)v_3\left(x\right)\dots +(-1{)}^{n}n!v_{n+1}(x)$

$P\left(x\right)v_1\left(x\right)+P^{\mathrm{\prime }}\left(x\right)v_2\left(x\right)+P^{\prime \prime }\left(x\right)v_3\left(x\right)\dots +n{v}_{n+1}\left(x\right)$

$P\left(x\right)v_1\left(x\right)-\frac{P^{\mathrm{\prime }}\left(x\right)v_2\left(x\right)}{2!}+\frac{P^{\prime \prime }\left(x\right)v_3\left(x\right)}{3!}\dots +(-1{)}^n\frac{v_{n+1}\left(x\right)}{(n+1)!}$