Use trigonometric identity before integration

A question is this type if and only if it explicitly requires proving or using a trigonometric identity (like cos²x = (1+cos2x)/2 or tan²x = sec²x - 1) to enable integration.

28 questions · Standard +0.2

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OCR C4 2016 June Q2
5 marks Standard +0.3
2 Use integration to find the exact value of \(\int _ { \frac { 1 } { 16 } \pi } ^ { \frac { 1 } { 8 } \pi } \left( 9 - 6 \cos ^ { 2 } 4 x \right) \mathrm { d } x\).
OCR C4 2008 January Q7
8 marks Standard +0.3
  1. Given that $$A ( \sin \theta + \cos \theta ) + B ( \cos \theta - \sin \theta ) \equiv 4 \sin \theta$$ find the values of the constants \(A\) and \(B\).
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 4 \sin \theta } { \sin \theta + \cos \theta } \mathrm { d } \theta$$ giving your answer in the form \(a \pi - \ln b\).
AQA C3 2007 June Q8
12 marks Standard +0.3
8
  1. Write down \(\int \sec ^ { 2 } x \mathrm {~d} x\).
  2. Given that \(y = \frac { \cos x } { \sin x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
  3. Prove the identity \(( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x\).
  4. Hence find \(\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x\), giving your answer to two significant figures.