Integration by parts done fast
Even though I've been through countless instances where I needed to use integration by parts, to this day, I still derive it from the chain rule, identify my 'parts' and apply the formula.
At some lecture a while back, one of my profs did a two step integration by parts in like 30 seconds. It went so fast, looked so easy, and I never caught on to how he did it. The answer just fell out.
What are you favorite tricks for quick integration by parts? Everything goes, mnemonics, special notation, or maybe some other trickery?
Speed is the number one priority here. I'd rather have a fast method that requires some practice to master, than have a slow method that can be learnt quickly!
Thanks in advance!
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$\begingroup$The main result is $$\int udv = uv - \int vdu$$ which means you pick out 2 parts from your integral -- you will differentiate one, and integrate the other one, which often will give you some idea of what the next integral will look like before any arithmetic was carried out.
$\endgroup$ 2 $\begingroup$It might have been the Stand and Deliver "tic-tac-toe" method.
It involves a table: derivatives of one function, anti-derivatives of the other function, and a column with an alternating sign.
So things like $\int x^5 \cos(x) dx$ are really fast. Take derivatives of $x^5 (5x^4, 20x^3, 60x^2, 120x, 120)$, anti-derivatives of $\cos(x) [\sin(x), -\cos(x), -\sin(x), \cos(x), \sin(x), -\cos(x)]$. Multiply the $x^5$ by the first anti-derivative in the list, and keep the sign ($x^5 \sin(x)$). Multiply the $5x^4$ by the second anti-derivative, and switch the sign ($5x^4 \cos(x)$). Continue until the term is zero:
$$\int x^5 \cos(x) dx \\ = x^5 \sin(x)+5 x^4 \cos(x)-20 x^3 \sin(x)-60 x^2 \cos(x)+120 x \sin(x)+120 \cos(x)+C.$$
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