Integration Using Polynomial Division

Questions that require performing polynomial division first, then integrating the resulting expression (quotient plus remainder term), often giving answers in the form a + ln(b).

10 questions · Standard +0.3

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CAIE P2 2020 June Q7
12 marks Standard +0.3
7
  1. Find the quotient when \(9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1\) is divided by ( \(3 x + 2\) ), and show that the remainder is 9 .
  2. Hence find \(\int _ { 1 } ^ { 6 } \frac { 9 x ^ { 3 } - 6 x ^ { 2 } - 20 x + 1 } { 3 x + 2 } \mathrm {~d} x\), giving the answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers.
  3. Find the exact root of the equation \(9 e ^ { 9 y } - 6 e ^ { 6 y } - 20 e ^ { 3 y } - 8 = 0\).
    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 2024 June Q5
8 marks Standard +0.3
5 The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } + 5 x + 4\).
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 3 x + 2 )\), and show that the remainder is 6 . \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-08_2713_33_146_2012} \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-09_2723_33_138_20}
  2. Find the value of \(\int _ { 0 } ^ { 2 } \frac { \mathrm { p } ( x ) } { 3 x + 2 } \mathrm {~d} x\) ,giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers.
CAIE P2 2022 November Q6
9 marks Standard +0.3
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 12 x ^ { 3 } - 9 x ^ { 2 } + 8 x - 4$$
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 4 x - 3 )\) and show that the remainder is 2 .
  2. Hence find \(\int _ { 2 } ^ { 12 } \left( \frac { \mathrm { p } ( x ) } { 4 x - 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\), giving your answer in the form \(a + \ln b\).
CAIE P2 2019 June Q5
8 marks Standard +0.3
5
  1. Find the quotient and remainder when \(2 x ^ { 3 } + x ^ { 2 } - 8 x\) is divided by ( \(2 x + 1\) ).
  2. Hence find the exact value of \(\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x\), giving the answer in the form \(\ln \left( k \mathrm { e } ^ { a } \right)\) where \(k\) and \(a\) are constants.
CAIE P3 2020 June Q5
8 marks Standard +0.3
5
  1. Find the quotient and remainder when \(2 x ^ { 3 } - x ^ { 2 } + 6 x + 3\) is divided by \(x ^ { 2 } + 3\).
  2. Using your answer to part (a), find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } - x ^ { 2 } + 6 x + 3 } { x ^ { 2 } + 3 } \mathrm {~d} x\).
CAIE P3 2022 March Q8
8 marks Standard +0.3
8
  1. Find the quotient and remainder when \(8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7\) is divided by \(4 x ^ { 2 } + 1\).
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 8 x ^ { 3 } + 4 x ^ { 2 } + 2 x + 7 } { 4 x ^ { 2 } + 1 } \mathrm {~d} x\).
CAIE P3 2024 November Q9
8 marks Standard +0.3
9
  1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
  2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).
Pre-U Pre-U 9794/1 2013 November Q12
Standard +0.3
12 The diagram shows the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) for \(x > - 1\). \includegraphics[max width=\textwidth, alt={}, center]{806dc286-416e-4785-8d13-0d524f808cb0-3_435_874_897_639}
  1. Find the coordinates of the points where the curve crosses the axes.
  2. Express \(\frac { x ^ { 2 } - 3 } { x + 1 }\) in the form \(A x + B + \frac { C } { x + 1 }\), where \(A , B\) and \(C\) are constants, and hence show that the exact area enclosed by the \(x\)-axis, the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) and the lines \(x = 2\) and \(x = 4\) is \(4 + \ln \frac { 9 } { 25 }\).
CAIE P2 2024 June Q5
8 marks Standard +0.3
The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).
  1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
  2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]
AQA C4 2016 June Q3
8 marks Standard +0.3
  1. Express \(\frac{3 + 13x - 6x^2}{2x - 3}\) in the form \(Ax + B + \frac{C}{2x - 3}\). [4 marks]
  2. Show that \(\int_3^6 \frac{3 + 13x - 6x^2}{2x - 3} \, dx = p + q \ln 3\), where \(p\) and \(q\) are rational numbers. [4 marks]