Factorization then partial fractions

First show that a polynomial has a specific factor or factorize a cubic/higher polynomial, then use partial fractions to integrate a related rational function.

2 questions · Challenging +1.1

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CAIE P3 2015 November Q7
10 marks Standard +0.3
7
  1. Show that ( \(x + 1\) ) is a factor of \(4 x ^ { 3 } - x ^ { 2 } - 11 x - 6\).
  2. Find \(\int \frac { 4 x ^ { 2 } + 9 x - 1 } { 4 x ^ { 3 } - x ^ { 2 } - 11 x - 6 } \mathrm {~d} x\).
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]