Category:Primitive of Power of x by Exponential of a x

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This category contains pages concerning Primitive of $x^n e^{a x}$:

Let $n$ be a positive integer.

Let $a$ be a non-zero real number.


\(\ds \int x^n e^{a x} \rd x\) \(=\) \(\ds \frac {e^{a x} } a \paren {x^n - \dfrac {n x^{n - 1} } a + \dfrac {n \paren {n - 1} x^{n - 2} } {a^2} - \dfrac {n \paren {n - 1} \paren {n - 2} x^{n - 3} } {a^3} + \cdots + \dfrac {\paren {-1}^n n!} {a^n} } + C\)
\(\ds \) \(=\) \(\ds \frac {e^{a x} } a \sum_{k \mathop = 0}^n \paren {\paren {-1}^k \frac {n^{\underline k} x^{n - k} } {a^k} } + C\)

where $n^{\underline k}$ denotes the $k$th falling factorial power of $n$.


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