In addition to the common integration shortcut of $\int x \cdot e^{ax}=\frac{xe^{ax}}{a}-\frac {e^{ax}}{a^2}+c$
Is there supplemental shortcut that can help me to avoid tedious repetitive integration by parts for when I do
$\int x^n\cdot e^{ax}dx$ ?
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$\begingroup$Do enough integrals of the form $\int x^ne^x\,dx$ and you'll notice the answer is $e^x$ times a polynomial of degree $n$. So you could cheat and guess a solution of this form, differentiate it, and match coefficients.
For example, guess$$\int x^3e^x\,dx=(ax^3+bx^2+cx+d)e^x + C. $$Differentiate, using the product rule on the RHS:$$ x^3e^x=[(ax^3+bx^2+cx+d) + (3ax^2+2bx+c)]e^x $$Match coefficients on $x^3, x^2, \dots$ on down and read off the answer: $$\begin{aligned}a&=1\\b&=-3a=-3\\c&=-2b=6\\d&=-c=-6\end{aligned}$$(Notice the pattern $-3, -2, -1$ !)
For integrals like $\int x^ne^{ax}\,dx$, reduce to this simpler case by substituting $u:=ax$.
$\endgroup$ $\begingroup$You begin by putting$$t=ax$$
and for $n\ge 0$, you compute
$$I_n=\int t^ne^tdt$$
As pointed by @Elliot,
$$I_{n+1}=-(n+1)I_n+ t^{n+1}e^t$$
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