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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

$\int x^n\cdot e^{ax}dx$ ?

PortMadeleineCrumpet
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    I'm not sure, but maybe tabular integration would help if you haven't heard of it? http://ramanujan.math.trinity.edu/rdaileda/teach/s18/m3357/parts.pdf – teddy May 22 '20 at 21:56
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    Find the pattern once, prove it using induction, and you'll never have to do it again. – pancini May 22 '20 at 21:58
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    This page has the formula for $\int x^n e^{x}, dx$: https://math.stackexchange.com/questions/46469/how-to-integrate-int-xn-ex-dx. You can use this to get the formula for $\int x^n e^{ax}, dx$ (e.g. substitute $u=ax$ in your integral first, so you'll find you just need to multiply the formula in that post by a certain constant involving $a$ and $n$). – Minus One-Twelfth May 22 '20 at 22:11
  • Does this answer your question? How to integrate $ \int x^n e^x dx$? – KReiser May 23 '20 at 04:11

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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 $u:=ax$.

grand_chat
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You begin by putting $$t=ax$$

and for $n\ge 0$, you compute

$$I_n=\int t^ne^tdt$$

As pointed by @Elliot,

$$I_{n+1}=-(n+1)I_n+ t^{n+1}e^t$$

hamam_Abdallah
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