My student frowns as I write down $u = \cos(x)$. “Wait wait wait”, he says, “how do you know to do a substitution?”

The honest answer is, I just do. I’ve done integration sum after integration sum for the last 20 years and it comes naturally. However, I know that’s not the answer he wants to hear. It’s time to come up with a process for him.

And I think: ‘aha! that’ll make for a good blog post on Wednesday.’

(Caveat: this article is based on what you need to know for Edexcel C4. It might work for other modules too.)

The last time I counted, there were six C4 integration methods:

  • By inspection

  • By partial fractions

  • Parametrically

  • By trig identity

  • By substitution (including function-derivative)

  • By parts

The process

The first thing you think, when you see an integral, is ‘can I integrate it as it stands?’ Is it a nice, simple integral, maybe a bunch of $x^n$s all added together? A $\cos(x)$ or another function with simple arguments? If so, then have at it! Integrate away.

Unfortunately, you don’t see many of those in C4, so you need to keep your eyes out for other possibilities.

The next most obvious ones are partial fractions and parametric integration - I’m not telling you how to do those here, just how to spot them, and that’s almost too obvious to say: if you’re doing the partial fractions question (or anything that involves a fraction with a factorisable bottom) and it asks you to integrate, you use partial fractions. If you’re doing the parametric question and it asks you to integrate, you do it parametrically. Easy so far?

The other easy one to spot is the trigonometric identity. Whenever you see a $\cos^2(x)$, or $/sin^2(x)$, you’re going to use an identity for $\cos(2x)$. If you have a $\tan^2(x)$ or a $\cot^2(x)$, you’ll use an identity to turn it into $\sec^2(x)$ or $\cosec^2(x)$ - both of which are in the formula book. Easy!

The big two

More commonly, though, you need to pick between a substitution and parts. Here’s how:

  • If you’re told to do it by parts or using a particular substitution, why have you even read this far? Do what they tell you.

  • If there’s an ugly bracket or bottom of a fraction, try the substitution $u =$ the ugly thing.

  • If there’s a $\ln(x)$ or similar knocking around, it almost certainly by parts - let $u$ be the logarithm

  • If there’s an $x^n$ multiplied by a function, try parts - let u be $x^n$ and hope that it goes away quickly! You might need to do this several times

  • If all else fails, try something. It’s better than leaving it blank.

So there you go: a quick guide to how to pick an integration method. Did I miss anything?