Partial fractions then inverse trig integration

A question is this type if and only if it requires decomposing into partial fractions first, then integrating terms that yield inverse trigonometric functions.

5 questions · Challenging +1.2

1.02y Partial fractions: decompose rational functions
Sort by: Default | Easiest first | Hardest first
OCR Further Pure Core 1 2021 June Q5
6 marks Challenging +1.2
5
Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.
OCR FP2 2009 January Q9
12 marks Standard +0.8
A curve has equation $$y = \frac{4x - 3a}{2(x^2 + a^2)},$$ where \(a\) is a positive constant.
  1. Explain why the curve has no asymptotes parallel to the \(y\)-axis. [2]
  2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve. [5]
  3. Find the exact value of \(\int_a^{2a} \frac{4x - 3a}{2(x^2 + a^2)} dx\), showing that it is independent of \(a\). [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 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]
SPS SPS FM Pure 2024 February Q14
6 marks Challenging +1.8
Show that \(\int_0^{\frac{1}{\sqrt{3}}} \frac{4}{1-x^4} dx = \ln(a + \sqrt{b}) + \frac{\pi}{c}\) where \(a\), \(b\) and \(c\) are integers to be determined. [6]