Multi-part with preliminary simplification

A question is this type if and only if it requires first simplifying an expression using trigonometric identities, algebraic manipulation, or other techniques in an early part, then applying integration by parts to the simplified form in a subsequent part.

3 questions · Standard +0.3

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Edexcel C34 2016 January Q8
9 marks Standard +0.3
8. $$f ( \theta ) = 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta$$
  1. Show that \(\mathrm { f } ( \theta ) = a + b \cos 2 \theta\), where \(a\) and \(b\) are integers which should be found.
  2. Using your answer to part (a) and integration by parts, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta ^ { 2 } \mathrm { f } ( \theta ) \mathrm { d } \theta$$
Edexcel C4 2010 June Q6
10 marks Standard +0.3
6. $$f ( \theta ) = 4 \cos ^ { 2 } \theta - 3 \sin ^ { 2 } \theta$$
  1. Show that \(f ( \theta ) = \frac { 1 } { 2 } + \frac { 7 } { 2 } \cos 2 \theta\).
  2. Hence, using calculus, find the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta \mathrm { f } ( \theta ) \mathrm { d } \theta\).
OCR C4 2006 June Q8
9 marks Standard +0.3
8
  1. Show that \(\int \cos ^ { 2 } 6 x \mathrm {~d} x = \frac { 1 } { 2 } x + \frac { 1 } { 24 } \sin 12 x + c\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } x \cos ^ { 2 } 6 x \mathrm {~d} x\).