Please, I need find a reduced formula to $\int x^m \cdot e^x dx$ by parts integration method, I've found that every time I compute the m is reduced so: $m-1$, $m-2$, $m-3$, but I couldn't found the general formula.
Thanks so much.
Please, I need find a reduced formula to $\int x^m \cdot e^x dx$ by parts integration method, I've found that every time I compute the m is reduced so: $m-1$, $m-2$, $m-3$, but I couldn't found the general formula.
Thanks so much.
$I_m = \int x^m e^x dx = \int x^m d(e^x) = x^m e^x - \int m x^{m-1} e^x dx= x^m e^x - mI_{m-1} + c_{m-1}$ \begin{align} I_m & = x^m e^x - m \left(x^{m-1} e^x - (m-1) I_{m-2} \right) + c_{m-2}\\ & = x^m e^x - m x^{m-1} e^x + m(m-1) I_{m-2} + c_{m-2}\\ & = x^m e^x - m x^{m-1} e^x + m(m-1) x^{m-2} e^x - m(m-1)(m-2)I_{m-3} + c_{m-3}\\ & = e^x \left(x^m - m x^{m-1} + m(m-1) x^{m-2} - \cdots + (-1)^m m!\right) + c_{0}\\ & = e^x \left(\sum_{k=0}^m P(m,k)(-1)^k x^{m-k}\right) + c_0\\ & = e^x \left(\sum_{k=0}^m P(m,k)(-1)^k x^{m-k}\right) + c_0\\ & = (-1)^m \Gamma(m+1,-x) + c_0 \end{align} where $P(m,k) = \dfrac{m!}{(m-k)!}$.