Integration by Substitution

Current UK exam textbooks pass over proofs and mathematical discussions in a hurry to show the ‘how to’ of exam questions.

Integration by substitution is a little more than just backwards chain-rule and deserves a fuller treatment.

Try this,


y=\displaystyle \int \textnormal{f}(x) \ \textnormal{d}x



Suppose that there exists a function g, of another variable u, such that x=\textnormal{g}(u) and let, \textnormal{f}(x)=\textnormal{f}(\textnormal{g}(u))=\textnormal{F}(u). So that,


Now, by the chain rule,

\dfrac{\textnormal{d}y}{\textnormal{d}u}=\dfrac{\textnormal{d}y}{\textnormal{d}x}\times \dfrac{\textnormal{d}x}{\textnormal{d}u}=\textnormal{F}(u)\dfrac{\textnormal{d}x}{\textnormal{d}u}


y=\displaystyle \int \dfrac{\textnormal{d}y}{\textnormal{d}u} \ \textnormal{d}u=\displaystyle \int \textnormal{F}(u)\dfrac{\textnormal{d}x}{\textnormal{d}u} \ \textnormal{d}u


y=\displaystyle \int \textnormal{f}(x) \ \textnormal{d}x=\displaystyle \int \textnormal{F}(u)\dfrac{\textnormal{d}x}{\textnormal{d}u} \ \textnormal{d}u

The linear combination of a sine and a cosine is itself a sine wave

A linear combination of two functions, f(x) and g(x) is a sum involving constant multiples of the functions.  That is,

a f(x)+b g(x)

where a, b \in \mathbb{R}.

So, in the case of \sin x and \cos x, we would have,


It is a slightly surprising fact that the linear combination of two sine waves is itself a sine wave.  The set of sine waves is closed under linear combination.

The in the featured animation, a\sin(x) is green and dotted and b\cos(x) is red and dotted.  The resulting linear combination is the continuous blue line.  The value a is set to 2.5, whereas the value b is animated.

This principle occurs in A level maths, Core 3, and is responsible for many long and complex questions.

support for school mathematics exams

I have developed a database driven website which shares past papers, markschemes and my own ‘write outs’ for most Edexcel modules in mathematics and further mathematics:

jped Exam Bucket

I use this mostly as a resource for my own teaching; using it I can lay my hands of exam questions and solutions of different types very quickly.  My students and pupils find it useful too because it shows how to do exam questions using the techniques taught in class; there is an important difference with mark schemes here.

The UK mathematics exam system is in a state of flux at the moment.  I see this is as a great opportunity to review and refresh all my teaching material.  I look forward to generating new material to support a revised school exam system.