## Substitution of u=tan(x/2)

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

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 Recognitions: Gold Member Homework Help Science Advisor It is useful when your integrand consists of a ratio between to polynomials in of trignometric functions. For example: Let's look at: $$\int\frac{\cos\theta}{2\cos\theta-\sin\theta}d\theta$$ How would you integrate that one? Not very easy, but look at the following: $$\cos\theta=\cos^{2}\frac{\theta}{2}-\sin^{2}\frac{\theta}{2}=\cos^{2}\frac{\theta}{2}(1-\tan^{2}\frac{\theta}{2})=\frac{1-u^{2}}{1+u^{2}}$$ $$\sin\theta=2\sin\frac{\theta}{2}\cos\frac{\theta}{2}=2\cos^{2}\frac{\th eta}{2}\tan\frac{\theta}{2}=\frac{2u}{1+u^{2}}$$ Thus, we also have: $$\tan\theta=\frac{2u}{1-u^{2}}$$ In addition, we have: $$\frac{du}{d\theta}=\frac{1}{2}\frac{1}{\cos^{2}\frac{\theta}{2}}=\frac{ 1}{2}(u^{2}+1}\to{d\theta}=\frac{2du}{1+u^{2}}$$ 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.