How The Mathematical Pirate Integrates By Parts

The student sighed. "$\int x^3 e^{-2x} \dx$", he said. "That's going to be integration by parts. And it's going to take three steps. What a pain."

"Aharr!" swashbuckled the Mathematical Pirate. "That's what you think!"

"It is what I think," said the student, slightly bemused.

"Would you like to know..." The Mathematical Pirate sidled up closer "... the secret pirate method?"

"I dunno," said the student. "Is it quicker?"

"Quicker?!" snorted the Mathematical Pirate. "Quicker than a schooner in a storm."

"I presume that's quick."

"... yes."

"OK, then, bring it."

"You're happy deciding what to differentiate and what to integrate?"

The student rolled his eyes. "$x^3$ gets simpler if you differentiate, so you differentiate that."

"Very good. So, in your first column, column the first, you write the $x^3$ and its derivatives - but alternating signs."

"$x^3$, $-3x^2$, $6x$ and $-6$. And the second column must be something to do with the integrals of $e^{-2x}$"

"Arr, exactly that."

"$e^{-2x}$, $-\frac{1}{2}e^{-2x}$, $\frac{1}{4}e^{-2x}$, $-\frac{1}{8}e^{-2x}$."

"One more."

"One more? $\frac{1}{16}e^{-2x}$. Molest me not with this pocket calculator stuff."

"Now multiply each thing in the left column by the thing below it and to the right."

"Like this?"

  Derivatives Integrals Products
Original $x^3$ $e^{-2x}$  
First $-3x^2$ $-\frac{1}{2}e^{-2x}$ $-\frac{x^3}{2}e^{-2x}$
Second $6x$ $\frac{1}{4}e^{-2x}$ $-\frac{3x^2}{4}e^{-2x}$
Third $-6$ $-\frac{1}{8}e^{-2x}$ $-\frac{3x}{4}e^{-2x}$
Fourth   $\frac{1}{16}e^{-2x}$ $-\frac{3}{8}e^{-2x}$

"Just like that, me hearty. Now add up the products."

"Can I factor out, say, $-\frac{1}{8}e^{-2x}$ first?"

The Mathematical Pirate's eyes narrowed. "Have you been talking with the Ninja again?"

"Maybe," said the student. "So, $-\frac{1}{8}e^{-2x}$ outside the bracket, then $4x^3 + 6x^2 + 6x + 3$?"

"Plus a constant," muttered the Mathematical Pirate, in the same tone as a parrot might say "Pieces of eight."

* Edited 2016-11-28 to fix a sign error. Many thanks to @seanelvidge for pointing it out.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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