微积分

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微积分化简

定义下面函数:

\[f(n,a)=\int x^ne^{ax}dx\]

由于

\[(x^ne^{ax})'=nx^{n-1}e^{ax}+ax^ne^x\]

因此

\[f(n)\\ =\int x^ne^{ax}dx\\ =\frac{1}{a}\int ((x^ne^{ax})'-nx^{n-1}e^{ax})dx\\ =\frac{1}{a}(x^ne^{ax}-n\int x^{n-1}e^{ax}dx)\\ =\frac{1}{a}(x^ne^{ax}-nf(n-1,a))\\\]

下面给出f(n)的具体递推公式:

\[f(n,a)= \left\{ \begin{array}{} \frac{1}{a}e^{ax} &,n=0\\ \frac{1}{a}(x^ne^{ax}-nf(n-1,a))&,n\geq 0 \end{array} \right.\]