# March, 2016

## Ask Uncle Colin: Integration rules

Dear Uncle Colin, Why can’t I work out $\int \left( \ln(x) \right)^2 \dx$ using the reverse chain rule? — Previously Acceptable, Reasonable Technique Stumbles Hello, PARTS, There are two answers to this: the first is, you can’t use the reverse chain rule — which I learned as ‘function-derivative’ when I

## Further Decimal Curiosities

This post is based on work by Mark Ritchings; I know of no finer1 maths tutor in Bury. A few weeks ago, I pointed in the vague direction of a few decimal curiosities — fractions that spit out lovely patterns in their decimal expansions. Having found one that generated the

## Ask Uncle Colin: rearranging $\cos^3(x)$

Dear Uncle Colin, I recently came across a problem in which I had to integrate $\cos^3(x)$. Somewhere in my mind, I recall that the thing to do is to make it into something involving $\cos(3x)$, but I couldn’t put the details together. Could you help? — Not A Very Inspired

## Review: Is This Prime?

Is This Prime? is probably the most infuriating, addictive, revolting and unbearably simple games that has ever disgraced my computer screen. I love it, hate it, am glad of its existence and wish it had never been written. It’s pretty tough to think of a simpler premise: you’re given an

## Ask Uncle Colin: A STEP integral

Dear Uncle Colin, I was doing a STEP paper and it asked me to calculate $\int_0^1 x^3 \arctan\left(\frac{1-x}{1+x}\right) \dx$, given that $\int_0^1 \frac{x^4}{1+x^2} \dx = \frac{\pi}{4} – \frac{2}{3}$. Nut-uh. College Asked Me Back: Rocked Interview. Daren’t Get Excited Hello, CAMBRIDGE! Is this thing on? Even with the given hint, this

## The Mathematical Ninja finds the value of $\pi$

This article has also been published as part of the Relatively Prime zine. The student yelped, and found his wrists and ankles strapped to the wheel before the lesson had even started. “Good morning,” said the Mathematical Ninja. “I think you and I need to have a little… talk.” The

## Ask Uncle Colin: Changing the variable (FP2 Differential equations)

Dear Uncle Colin, I’m supposed to use the change of variable $z = \sin(x)$ to turn $\cos(x) \diffn{2}{y}{x} + \sin(x) \dydx – 2y \cos^3(x)= 2\cos^5(x)$ into $\diffn{2}{y}{z} – 2y = 2\left(1-z^2\right)$. Yeah but no but. No idea. Lacking, Obviously, Something Trivial Hello, LOST, Right, yes. Nasty one, this. The main

## Second derivatives, reciprocals, and the chain rule

Once in a while, a student puts me on the spot; it’s not always deliberate. In this case, changing a variable in a second-order differential equation, he blithely said “Well, $\diffn 2yx = \frac{1}{\diffn 2xy}$…” “Whoa whoa whoa…” “… isn’t it?” Now, had I not had a series of discussions

## Wrong, But Useful: Episode 33

In Episode $n$ of Wrong, But Useful (where $n$ is a semiprime followed by two semiprimes and a square), @reflectivemaths and I discuss: Colin’s trip to Center Parcs and whether it is sufficiently central The number of the podcast, which is the tetrahedral angle, $\arccos\left(\frac{1}{3}\right)$ The Other Half podcast Our

## Ask Uncle Colin: Why does the finger trick work?

Dear Uncle Colin, In Basic Maths For Dummies, you mention a method for multiplying numbers from 6 × 6 to 10 × 10 on your fingers. It’s almost magical! Why does it work? — Does It Guarantee Interesting Times Sums? Hello, DIGITS, I’m glad you’re finding the book helpful! The