8.06a Reduction formulae: establish, use, and evaluate recursively

107 questions

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CAIE FP1 2019 November Q3
7 marks Challenging +1.3
The integral \(I_n\), where \(n\) is a positive integer, is defined by $$I_n = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^n \sin \pi x \, dx.$$
  1. Show that $$n(n+1)I_{n+2} = 2^{n+1}n + \pi - \pi^2 I_n.$$ [5]
  2. Find \(I_5\) in terms of \(\pi\) and \(I_1\). [2]
CAIE Further Paper 2 2020 June Q2
6 marks Challenging +1.2
Let \(I_n = \int_0^1 (1+3x)^n e^{-3x} dx\), where \(n\) is an integer.
  1. Show that \(3I_n = 1 - 4^n e^{-3} + 3nI_{n-1}\). [3]
  2. Find the exact value of \(I_2\). [3]
CAIE Further Paper 2 2021 November Q8
13 marks Challenging +1.2
  1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh^2 x = \sech^2 x.$$ [3]
  2. Using the substitution \(u = \tanh x\), or otherwise, find \(\int \sech^2 x \tanh^2 x \, dx\). [2]
  3. It is given that, for \(n \geq 0\), \(I_n = \int_0^{\ln 3} \sech^n x \tanh^2 x \, dx\). Show that, for \(n \geq 2\), $$(n + 1)I_n = \left(\frac{4}{3}\right)^{\frac{3}{n-2}} + (n - 2)I_{n-2}.$$ [You may use the result that \(\frac{d}{dx}(\sech x) = -\tanh x \sech x\).] [5]
  4. Find the value of \(I_4\). [3]
OCR Further Additional Pure 2017 Specimen Q5
9 marks Challenging +1.2
In this question you must show detailed reasoning. It is given that \(I_n = \int_0^\pi \sin^n \theta \, d\theta\) for \(n \geq 0\).
  1. Prove that \(I_n = \frac{n-1}{n} I_{n-2}\) for \(n \geq 2\). [5]
  2. Evaluate \(I_1\) and use the reduction formula to determine the exact value of \(\int_0^\pi \cos^2 \theta \sin^5 \theta \, d\theta\). [4]
Pre-U Pre-U 9795/1 2013 November Q13
24 marks Hard +2.3
  1. Let \(I_n = \int_0^{\alpha} \cosh^n x \, dx\) for integers \(n \geqslant 0\), where \(\alpha = \ln 2\).
    1. Prove that, for \(n \geqslant 2\), \(nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}\). [5]
    2. A curve has parametric equations \(x = 12 \sinh t + 4 \sinh^3 t\), \(y = 3 \cosh^4 t\), \(0 \leqslant t \leqslant \ln 2\). Find the length of the arc of this curve, giving your answer in the form \(a + b \ln 2\) for rational numbers \(a\) and \(b\). [8]
  2. The circle with equation \(x^2 + (y - R)^2 = r^2\), where \(r < R\), is rotated through one revolution about the \(x\)-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of \(\pi\), \(R\) and \(r\), the surface area of this torus. [11]
Pre-U Pre-U 9795/1 2015 June Q12
22 marks Challenging +1.8
Let \(I_n = \int_0^2 x^n \sqrt{1 + 2x^2} \, \text{d}x\) for \(n = 0, 1, 2, 3, \ldots\).
    1. Evaluate \(I_1\). [3]
    2. Prove that, for \(n \geqslant 2\), $$(2n + 4)I_n = 27 \times 2^{n-1} - (n - 1)I_{n-2}.$$ [6]
    3. Using a suitable substitution, or otherwise, show that $$I_0 = 3 + \frac{1}{\sqrt{2}} \ln(1 + \sqrt{2}).$$ [8]
  2. The curve \(y = \frac{1}{\sqrt{2}} x^2\), between \(x = 0\) and \(x = 2\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface with area \(S\). Find the exact value of \(S\). [5]
Pre-U Pre-U 9795 Specimen Q14
13 marks Challenging +1.8
Let \(J_n = \int_1^{\mathrm{e}} (\ln x)^n \, \mathrm{d}x\), where \(n\) is a positive integer. By considering \(\frac{\mathrm{d}}{\mathrm{d}x}(x(\ln x)^n)\), or otherwise, show that $$J_n = \mathrm{e} - nJ_{n-1}.$$ [4] Let \(J_n = \frac{J_n}{n!}\). Show that $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots + \frac{1}{10!} = \frac{1}{\mathrm{e}}(1 + J_{10}).$$ [6] It can be shown that $$\sum_{r=2}^{n} \frac{(-1)^r}{r!} = \frac{1}{\mathrm{e}}(1 + (-1)^n J_n)$$ for all positive integers \(n\). Deduce the sum to infinity of the series $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots,$$ justifying your conclusion carefully. [3]