Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions – and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can.

Dear Uncle Colin,

I’ve been trying to integrate $\int \frac{x^2}{x-1} \dx$ for what feels like days and I’m completely stuck. I worry that I’m not cut out for A-level!

-- Calculus: Amazingly Useful, Can Hurt You

Hi, there, CAUCHY – and don’t worry, not being able to do something doesn’t mean you’re not cut out for A-level. In fact, trying hard and then asking for help is exactly the kind of skill I try to foster in my students – so well done for having a stab.

With a question like this, there are (at least) two options: an easy one (the $u$-substitution) and a more involved one (the division method).

The $u$-substitution

The thing to ask yourself when contemplating a substitution is, ‘what’s the ugly thing here?’. Clearly enough, it’s the $x-1$ on the bottom of the fraction, so let’s say $u = x-1$ and – to howls of anguish from the purists, $\d u = \d x$.

Then our integral becomes $\int \frac{x^2}{u} \d u$. But wait! There’s still a nasty $x$ knocking around – fortunately, we can rearrange our substitution to say $x = u+1$, so the top can be written as $u^2 + 2u + 1$.

$\int \frac{u^2 + 2u + 1}{u} \d u = \int u + 2 + \frac 1u \d u$, which is much nicer. You get $\frac {u^2}{2} + 2u + \ln\left| u\right| + C$, nice as you like.

One last thing, though: we introduced the $u$ ourselves, so we need to turn it back into an $x$, giving

$\frac{(x-1)^2}{2} + 2(x-1) + \ln \left| x-1 \right| + C$.

(A smart-arse would tidy that up as $\frac {x^2}2 + x + \ln \left| x-1 \right| + c$. Can you see why that works?)

The division method

An alternative, which may be more comfortable, even if it’s longer-winded, is to simplify the fraction by division: $\frac{x^2}{x-1} = x+1 + \frac 1{x-1}$, which is almost trivial to integrate, giving $\frac {x^2} 2 + x + \ln \left| x-1 \right| + C$ – the same answer as the other way!

Hope that helps you find some rest!

-- Uncle Colin

* Edited 2015-07-22 to change a minus sign. Thanks, @realityminus3! * Edited 2016-05-18 to change brackets to absolute values. Thanks to Reuben for reminding me of the post!