1.08j Integration using partial fractions

142 questions

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OCR C4 Q5
10 marks Standard +0.3
$$f(x) = \frac{15-17x}{(2+x)(1-3x)^2}, \quad x \neq -2, \quad x \neq \frac{1}{3}.$$
  1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$f(x) = \frac{A}{2+x} + \frac{B}{1-3x} + \frac{C}{(1-3x)^2}.$$ [5]
  2. Find the value of $$\int_{-1}^{0} f(x) \, dx,$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers. [5]
OCR MEI C4 Q1
7 marks Moderate -0.3
Using partial fractions, find \(\int \frac{x}{(x+1)(2x+1)} dx\). [7]
OCR H240/03 2019 June Q5
9 marks Standard +0.3
In this question you must show detailed reasoning.
  1. Prove that \((\cot \theta + \cosec \theta)^2 = \frac{1 + \cos \theta}{1 - \cos \theta}\). [4]
  2. Hence solve, for \(0 < \theta < 2\pi\), \(3(\cot \theta + \cosec \theta)^2 = 2 \sec \theta\). [5]
AQA Paper 3 2019 June Q7
8 marks Standard +0.3
  1. Express \(\frac{4x + 3}{(x - 1)^2}\) in the form \(\frac{A}{x - 1} + \frac{B}{(x - 1)^2}\) [3 marks]
  2. Show that $$\int_3^4 \frac{4x + 3}{(x - 1)^2} \, dx = p + \ln q$$ where \(p\) and \(q\) are rational numbers. [5 marks]
AQA Paper 3 Specimen Q6
8 marks Challenging +1.2
Find the value of \(\int_1^2 \frac{6x + 1}{6x^2 - 7x + 2} dx\), expressing your answer in the form \(m\ln 2 + n\ln 3\), where \(m\) and \(n\) are integers. [8 marks]
WJEC Unit 3 2018 June Q5
8 marks Moderate -0.3
  1. Show that $$\frac{3x}{(x-1)(x-4)^2} = \frac{A}{(x-1)} + \frac{B}{(x-4)} + \frac{C}{(x-4)^2},$$ where \(A\), \(B\) and \(C\) are constants to be found. [3]
  2. Evaluate \(\int_5^7 \frac{3x}{(x-1)(x-4)^2} \, dx\), giving your answer correct to 3 decimal places. [5]
WJEC Further Unit 4 2019 June Q4
16 marks Standard +0.3
  1. Given that \(y = \cot^{-1} x\), show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-1}{x^2 + 1}\). [5]
  2. Express \(\frac{6x^2 - 10x - 9}{(2x + 3)(x^2 + 1)}\) in terms of partial fractions. [5]
  3. Hence find \(\int \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\). [5]
  4. Explain why \(\int_{-2}^{5} \frac{6x^2 - 8x - 6}{(2x + 3)(x^2 + 1)} \mathrm{d}x\) cannot be evaluated. [1]
WJEC Further Unit 4 2022 June Q14
10 marks Challenging +1.8
Evaluate the integral $$\int_2^4 \frac{6x^2 + 2x + 16}{x^3 - x^2 + 3x - 3} dx,$$ giving your answer correct to three decimal places. [10]
WJEC Further Unit 4 2023 June Q11
14 marks Challenging +1.2
Evaluate the integrals
  1. \(\int_{-2}^{0} e^{2x} \sinh x \, \mathrm{d}x\), [5]
  2. \(\int_{\frac{1}{2}}^{3} \frac{5}{(x-1)(x^2+9)} \, \mathrm{d}x\). [9]
WJEC Further Unit 4 2024 June Q5
14 marks Challenging +1.8
Find each of the following integrals.
  1. \(\int \frac{3-x}{x(x^2+1)} \mathrm{d}x\) [8]
  2. \(\int \frac{\sinh 2x}{\sqrt{\cosh^4 x - 9\cosh^2 x}} \mathrm{d}x\) [6]
WJEC Further Unit 4 Specimen Q7
10 marks Standard +0.8
The function \(f\) is defined by $$f(x) = \frac{8x^2 + x + 5}{(2x + 1)(x^2 + 3)}.$$
  1. Express \(f(x)\) in partial fractions. [4]
  2. Hence evaluate $$\int_2^5 f(x)dx,$$ giving your answer correct to three decimal places. [6]
SPS SPS FM 2021 April Q1
11 marks Moderate -0.3
  1. Differentiate the following with respect to \(x\), simplifying your answers fully
    1. \(y = e^{3x} + \ln 2x\) [1]
    2. \(y = (5 + x^2)^{\frac{3}{2}}\) [1]
    3. \(y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}\) [2]
    4. \(y = e^{-\frac{3}{x}} \ln(1 + x^3)\) [2]
  2. Integrate with respect to \(x\)
    1. \(\frac{7}{(2x-5)^8} - \frac{3}{2x-5}\) [2]
    2. \(\frac{4x^2+5x-3}{2x-5}\) [3]
SPS SPS SM Pure 2020 October Q6
5 marks Moderate -0.8
  1. Express \(\frac{x}{(x + 1)(x + 2)}\) in partial fractions. [3]
  2. Hence find \(\int \frac{x}{(x + 1)(x + 2)} dx\). [2]
SPS SPS SM 2021 November Q2
6 marks Moderate -0.3
  1. Express \(\frac{5x+7}{(x+3)(x+1)^2}\) in partial fractions. In this question you must show all of your algebraic steps clearly. [3] The function \(f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}\) can be written in the form; $$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
  2. Hence find the exact value of \(\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx\) [3]
OCR Further Pure Core 2 2018 March Q8
12 marks Challenging +1.8
In this question you must show detailed reasoning. Show that \(\int_0^2 \frac{2x^2 + 3x - 1}{x^3 - 3x^2 + 4x - 12} dx = \frac{3}{8}\pi - \ln 9\). [12]
OCR H240/02 2017 Specimen Q4
7 marks Moderate -0.3
  1. Express \(\frac{1}{(x-1)(x+2)}\) in partial fractions [2]
  2. In this question you must show detailed reasoning. Hence find \(\int_2^3 \frac{1}{(x-1)(x+2)} dx\). Give your answer in its simplest form. [5]
Pre-U Pre-U 9794/2 Specimen Q6
10 marks Standard +0.8
  1. Express \(y^3 - 3y - 2\) in terms of \(x\), where \(x = y + 1\). [1]
  2. Hence express $$\frac{2y + 5}{y^3 - 3y - 2}$$ in partial fractions. [5]
  3. Find the exact value of $$\int_0^1 \frac{2y + 5}{y^3 - 3y - 2} dy.$$ [4]