Integration using harmonic form

A question is this type if and only if it involves integrating an expression that requires expressing in harmonic form first, typically of the form 1/(a·sin + b·cos)².

7 questions · Standard +0.6

1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc
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CAIE P2 2021 March Q7
9 marks Standard +0.8
7
  1. Express \(5 \sqrt { 3 } \cos x + 5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  2. As \(x\) varies, find the least possible value of $$4 + 5 \sqrt { 3 } \cos x + 5 \sin x$$ and determine the corresponding value of \(x\) where \(- \pi < x < \pi\).
  3. Find \(\int \frac { 1 } { ( 5 \sqrt { 3 } \cos 3 \theta + 5 \sin 3 \theta ) ^ { 2 } } d \theta\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2007 June Q5
7 marks Standard +0.3
5
  1. Express \(\cos \theta + ( \sqrt { } 3 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + ( \sqrt { } 3 ) \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { 1 } { \sqrt { } 3 }\).
CAIE P3 2013 June Q9
10 marks Standard +0.3
9
  1. Express \(4 \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
  2. Hence
    1. solve the equation \(4 \cos \theta + 3 \sin \theta = 2\) for \(0 < \theta < 2 \pi\),
    2. find \(\int \frac { 50 } { ( 4 \cos \theta + 3 \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).
CAIE P3 2013 June Q4
7 marks Standard +0.3
4
  1. Express \(( \sqrt { } 3 ) \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( ( \sqrt { } 3 ) \cos x + \sin x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \sqrt { } 3$$
CAIE P2 2004 November Q8
10 marks Standard +0.3
8
  1. Express \(\cos \theta + \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that $$\frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } = \frac { 1 } { 2 } \sec ^ { 2 } \left( \theta - \frac { 1 } { 4 } \pi \right)$$
  3. By differentiating \(\frac { \sin x } { \cos x }\), show that if \(y = \tan x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x\).
  4. Using the results of parts (ii) and (iii), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = 1$$
OCR MEI C4 2009 January Q6
8 marks Standard +0.8
6
  1. Express \(\cos \theta + \sqrt { 3 } \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(\alpha\) is acute, expressing \(\alpha\) in terms of \(\pi\).
  2. Write down the derivative of \(\tan \theta\). Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { ( \cos \theta + \sqrt { 3 } \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { \sqrt { 3 } } { 4 }\).
CAIE P3 2024 June Q8
8 marks Challenging +1.2
  1. Express \(3 \cos 2x - \sqrt{3} \sin 2x\) in the form \(R \cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the exact values of \(R\) and \(\alpha\). [3]
  2. Hence find the exact value of \(\int_0^{\frac{1}{2}\pi} \frac{3}{(3 \cos 2x - \sqrt{3} \sin 2x)^2} \, dx\), simplifying your answer. [5]