Integration using reciprocal identities

A question is this type if and only if it requires proving an identity then using it to evaluate a definite or indefinite integral.

14 questions · Standard +0.4

1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)
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CAIE P2 2020 June Q8
10 marks Standard +0.3
8
  1. Show that \(3 \sin 2 \theta \cot \theta \equiv 6 \cos ^ { 2 } \theta\).
  2. Solve the equation \(3 \sin 2 \theta \cot \theta = 5\) for \(0 < \theta < \pi\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } 3 \sin x \cot \frac { 1 } { 2 } x \mathrm {~d} x\).
    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 P2 2021 June Q3
6 marks Standard +0.3
3
  1. Show that \(( \sec x + \cos x ) ^ { 2 }\) can be expressed as \(\sec ^ { 2 } x + a + b \cos 2 x\), where \(a\) and \(b\) are constants to be determined.
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \sec x + \cos x ) ^ { 2 } \mathrm {~d} x\).
CAIE P2 2023 November Q7
11 marks Standard +0.3
7
  1. Prove that \(\sin 2 x ( \cot x + 3 \tan x ) \equiv 4 - 2 \cos 2 x\).
  2. Hence find the exact value of \(\cot \frac { 1 } { 12 } \pi + 3 \tan \frac { 1 } { 12 } \pi\).
  3. \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-13_796_789_278_708} The diagram shows the curve with equation \(y = 4 - 2 \cos 2 x\), for \(0 < x < 2 \pi\). At the point \(A\), the gradient of the curve is 4 . The point \(B\) is a minimum point. The \(x\)-coordinates of \(A\) and \(B\) are \(a\) and \(b\) respectively. Show that \(\int _ { a } ^ { b } ( 4 - 2 \cos 2 x ) \mathrm { d } x = 3 \pi + 1\).
    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 P2 2020 Specimen Q7
11 marks Standard +0.3
7
  1. Show that \(\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }\) and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) \mathrm { d } x$$
  2. \includegraphics[max width=\textwidth, alt={}, center]{0af2714b-d3eb-4112-a869-eda5cf266cd8-13_535_771_274_648} The region enclosed by the curve \(y = \tan x + \cos x\) and the lines \(x = 0 , x = \frac { 1 } { 4 } \pi\) and \(y = 0\) is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the \(x\)-axis.
CAIE P3 2003 June Q10
10 marks Standard +0.3
10
  1. Prove the identity $$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x$$
  2. Show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \cot x \mathrm {~d} x = \frac { 1 } { 2 } \ln 2\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \operatorname { cosec } 2 x \mathrm {~d} x\), giving your answer in the form \(a \ln b\).
CAIE P3 2019 June Q3
7 marks Standard +0.3
3 Let \(f ( \theta ) = \frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta }\).
  1. Show that \(\mathrm { f } ( \theta ) = \tan \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).
CAIE P3 2012 November Q5
8 marks Standard +0.3
5
  1. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  2. Show that \(\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x\).
  3. Deduce that \(\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x\).
  4. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).
CAIE P3 2013 November Q5
7 marks Standard +0.3
5
  1. Prove that \(\cot \theta + \tan \theta \equiv 2 \operatorname { cosec } 2 \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln 3\).
CAIE P2 2008 November Q8
9 marks Standard +0.8
8
  1. (a) Prove the identity $$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 + \sin x } { \cos ^ { 2 } x }$$ (b) Hence prove that $$\sec ^ { 2 } x + \sec x \tan x \equiv \frac { 1 } { 1 - \sin x }$$
  2. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  3. Using the results of parts (i) and (ii), find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x$$
CAIE P3 2021 June Q4
6 marks Standard +0.3
4
  1. Prove that \(\frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \equiv \tan ^ { 2 } \theta\).
  2. Hence find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { 1 - \cos 2 \theta } { 1 + \cos 2 \theta } \mathrm {~d} \theta\).
CAIE P3 2021 June Q6
7 marks Standard +0.8
6
  1. Prove that \(\operatorname { cosec } 2 \theta - \cot 2 \theta \equiv \tan \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } ( \operatorname { cosec } 2 \theta - \cot 2 \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln 2\).
Edexcel C4 Q1
4 marks Standard +0.3
  1. Find
$$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$
CAIE P2 2014 June Q5
8 marks Standard +0.3
  1. Prove that \(\tan \theta + \cot \theta \equiv \frac { 2 } { \sin 2 \theta }\).
  2. Hence
    1. find the exact value of \(\tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 8 } \pi\),
    2. evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 6 } { \tan \theta + \cot \theta } \mathrm { d } \theta\).
Pre-U Pre-U 9794/2 2016 June Q9
11 marks Challenging +1.2
  1. Show that \(\frac{\sin x}{1 + \sin x} \equiv \sec x \tan x - \sec^2 x + 1\). [5]
  2. Hence show that \(\int_0^{\frac{\pi}{4}} \frac{\sin x}{1 + \sin x} \, dx = \frac{1}{4}\pi + \sqrt{2} - 2\). [6]