Hope this helps!Best regards,Nalin Pithwa

In summary, the substitution u=tan(x/2) can be used to simplify the integration of certain rational expressions involving trigonometric functions. It involves converting the integrand into a rational expression in terms of the variable u, and then using partial fractions decomposition to solve for the integral. This substitution used to be common, but is no longer part of the modern A-level syllabus.
  • #1
2^Oscar
45
0
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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  • #2
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{\theta}{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. :smile:
 

1. What is the substitution of u=tan(x/2)?

The substitution of u=tan(x/2) is a mathematical technique used to simplify and solve integrals containing trigonometric functions.

2. Why is the substitution of u=tan(x/2) useful?

The substitution of u=tan(x/2) is useful because it can help to simplify complex integrals and make them easier to solve. It also allows for the use of trigonometric identities to solve integrals.

3. How is the substitution of u=tan(x/2) performed?

To perform the substitution of u=tan(x/2), we replace all instances of the variable x with the expression u=tan(x/2). Then, we use the trigonometric identity cos^2(x/2) = (1+tan^2(x/2))/2 to simplify the integral.

4. Can the substitution of u=tan(x/2) be used for any integral containing trigonometric functions?

No, the substitution of u=tan(x/2) is only applicable to integrals that contain a trigonometric function of the form tan(ax+b) where a and b are constants.

5. Are there any limitations to using the substitution of u=tan(x/2)?

One limitation of using the substitution of u=tan(x/2) is that it only works for certain types of integrals and may not be applicable to all integrals containing trigonometric functions. Additionally, it may not always result in a simpler integral and may require further manipulations to solve.

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