Substitution of u=tan(x/2)

2^Oscar

Hi,

I've been doing some additional maths papers and I've seen the use of the substitution u=tan(x/2) in order to calculate integrals. In the mark scheme it states that this particular substitution used to be fairly common, however is not on the modern A-level syllabus.

Would someone please mind advising me of suitable situations to use such a substitution? I am struggling to see when I should use it.


Thanks,
Oscar

arildno

It is useful when your integrand consists of a ratio between to polynomials in of trignometric functions.

For example:

Let's look at:
[tex]\int\frac{\cos\theta}{2\cos\theta-\sin\theta}d\theta[/tex]

How would you integrate that one?

Not very easy, but look at the following:
[tex]\cos\theta=\cos^{2}\frac{\theta}{2}-\sin^{2}\frac{\theta}{2}=\cos^{2}\frac{\theta}{2}(1-\tan^{2}\frac{\theta}{2})=\frac{1-u^{2}}{1+u^{2}}[/tex]
[tex]\sin\theta=2\sin\frac{\theta}{2}\cos\frac{\theta}{2}=2\cos^{2}\frac{\th eta}{2}\tan\frac{\theta}{2}=\frac{2u}{1+u^{2}}[/tex]

Thus, we also have:
[tex]\tan\theta=\frac{2u}{1-u^{2}}[/tex]

In addition, we have:
[tex]\frac{du}{d\theta}=\frac{1}{2}\frac{1}{\cos^{2}\frac{\theta}{2}}=\frac{ 1}{2}(u^{2}+1}\to{d\theta}=\frac{2du}{1+u^{2}}[/tex]

Thus, the above integral can be converted into a rational expression of polynomials in the variable "u", and that can be solved using partial fractions decomposition.