Half-angle tangent substitution t = tan(x/2)

A question is this type if and only if it requires the Weierstrass substitution t = tan(x/2) to convert a trigonometric integral into a rational function.

3 questions · Challenging +1.2

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OCR FP2 2006 June Q5
7 marks Challenging +1.2
5
  1. Express \(t ^ { 2 } + t + 1\) in the form \(( t + a ) ^ { 2 } + b\).
  2. By using the substitution \(\tan \frac { 1 } { 2 } x = t\), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 + \sin x } \mathrm {~d} x = \frac { \sqrt { 3 } } { 9 } \pi$$
OCR FP2 2008 June Q3
6 marks
3 By using the substitution \(t = \tan \frac { 1 } { 2 } x\), find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 - \cos x } \mathrm {~d} x$$ giving the answer in terms of \(\pi\).
OCR FP2 2010 June Q3
6 marks Challenging +1.2
3 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x = 1 + \sqrt { 3 }$$