Integrating Into Integration

Don’t we love our integration slang.

[!Note] Some of this I use with my maths friends, some I use personally. Most of them aren’t widely established or accepted mathematical terms, but… you could help spread them ;)

catalyst

You’re looking for a catalyst here.

Multiplying through (or dividing by) a factor when integrating a fraction. Reference to Integration is Like Organic Chemistry.

conjugate

I’ll try multiplying through by the conjugate and see what happens.

In relation to an expression $p \pm q$ , “conjugate” refers to the expression with the opposite sign. This is extrapolated from the complex conjugate^↗.

For instance, the conjugate of $\sec{x}-\tan{x}$ is $\sec{x}+\tan{x}$ . Multiplying by the conjugate usually allows difference of 2 squares to be applied.

DBI method

dbi just use DBI method

A more cultured name for the DI method, a popular method for speedrunning integration by parts.

DOTS

Then use difference of 2 squares, so this factorises like that, and these factors cancel out!

Difference of 2 squares. Refers to the identity

x^2 - y^2 = (x-y)(x + y)

evolution

You probably only want one of primitives or evolutions.

Refers to $\tan$ and $\sec$ , which are ‘evolutions’ of the primitive trigonometric functions:

an{x} = rac{sin{x}}{cos{x}} qquad sec{x} = rac{1}{cos{x}}

Notes

$\cot$ and $\csc$ are also evolutions, ofc, but they feel a bit more ‘outlandish’.

We pair the trigonometric functions in this way because their derivatives match up nicely:

egin{align*} rac{d}{dx} sin{x} &= cos{x} qquad cos{x} = -sin{x} \ rac{d}{dx} an{x} &= sec^2{x} qquad sec{x} = sec{x} an{x} end{align*}

famously

e^x is famously both the derivative and antiderivative of e^x.

Well-known, well-established.

ICR

If you multiply through by $2$ , it’s easier to spot the inverse chain rule.

Inverse chain rule.

IPR

You can skip too the solution much faster here by spotting the inverse product rule.

Inverse product rule.

IQR

*It’s really, really hard to spot the inverse quotient rule in this integral.

Inverse quotient rule.

Notes

Famously not the inter-quartile range^↗.

JMWC

Oh my days it’s the JMWC himself.

Left as an exercise to the reader.

layer cake

This layer cake is really well-hidden.

Refers to integrating a fraction where the numerator is the derivative of the denominator, or an integer multiple of the derivative.

int rac{kf'(x)}{f(x)} dx = klnleft( f(x) ight) + c

If it is exactly the derivative, it becomes straight-up layer cake. Plain layer cake refers to the cases where the numerator is more disguised.

parts duplication

Woah, we got parts duplication!

A rare occurrence when using integration by parts, where the original integrand is obtained on the right side of the equation.

egin{align*} int e^x sin{x} &= e^x sin{x} - int e^x cos{x} dx \ &= e^x sin{x} - left( e^x cdot cos{x} - int e^x cdot -sin{x} dx ight) \ &= e^x sin{x} - e^x cdot cos{x} - extcolor{#4d9dcd}{int e^x cdot sin{x} dx} \ 2 int e^x sin{x} &= e^x sin{x} - e^x cdot cos{x} \ int e^x sin{x} &= rac{1}{2} left( e^x sin{x} - e^x cdot cos{x} ight) end{align*}

primitive

Try reducing to primitives.

Refers to $\sin$ and $\cos$ , the ‘primitive’ trigonometric functions. Can also be extended to mean $\sinh$ and $\cosh$ in a hyperbolic context.

See also evolution, which refers to $\tan$ and $\sec$ .

quotable

It’s quotable from here.

A solution is quotable if it’s sufficiently simple to jump straight to the answer, i.e. “quote” it. It can also refer to “quoting” a formula from a formula book.

int rac{1}{x^2 + 9} = rac{1}{3} an^{-1}left( rac{1}{3}x ight) + c

silver bullet

Don’t tell me I have to use silver bullet.

Alias for integration by parts. Reference to the fact that parts is famously “not a silver bullet”.

straight-up

Is this straight-up?

Yo, it’s just straight-up inverse product rule!

Often used as a shorthand for straight-up layer cake. Refers more generally to an integrand where one method can be applied exactly.

For instance, “straight-up inverse chain rule”:

10x^9 (x^10

Without the coefficient of $10$ it’s not “straight-up”; the $10x^9$ signposts quite obviously that it’s the derivative of $x^10$ .

straight-up layer cake

That’s straight-up layer cake, mate.

A degenerate case of layer cake where the numerator is the exact derivative of the denominator, in which case

int rac{f'(x)}{f(x)} dx = lnleft( f(x) ight) + c

tractorise

Starts with “f” and rhymes with tractorise.

Cockney rhyming slang(?) for factorise.

translation

Undo the translation and you find it’s odd.

Refers to an integral where the input has been translated, i.e. $x$ has been mapped to $x + a$ .

trivial

And the rest is trivial, happy days!

In the context of integration, this describes an integral that is sufficiently simplified such that we can be certain it is solvable – i.e. the method(s) or solution route is obvious. If we strip integration to its essence, then the goal when solving any integral is to reduce it to a trivial form.

Notes

It is important to clarify that “trivial” does not make any comments on the difficulty of carrying out the integration; it only describes the nature of the integral. A trivial integral may have an obvious method that is still nontrivial to execute accurately.

As is the case in the rest of mathematics, what one perceives as “trivial” is very much a function of one’s exposure to integration. To an adventurer who has seen

int rac{1}{xln{x}} dx

37 times before this is not a hard integral, but for someone who has only just started encountering integrals with $\ln(x)$ this could be very nontrivial to spot indeed.