Polynomial division before integration

A question is this type if and only if it requires dividing polynomials (or showing a quotient and remainder) before integrating the result.

4 questions · Standard +0.0

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CAIE P2 2023 November Q5
8 marks Moderate -0.3
5
  1. Find the quotient when \(6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4\) is divided by ( \(2 x + 1\) ), and show that the remainder is 6 .
  2. Hence find $$\int _ { 2 } ^ { 7 } \frac { 6 x ^ { 3 } - 5 x ^ { 2 } - 24 x - 4 } { 2 x + 1 } d x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are integers.
CAIE P2 2006 June Q7
11 marks Moderate -0.3
7
  1. Differentiate \(\ln ( 2 x + 3 )\).
  2. Hence, or otherwise, show that $$\int _ { - 1 } ^ { 3 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x = \ln 3$$
  3. Find the quotient and remainder when \(4 x ^ { 2 } + 8 x\) is divided by \(2 x + 3\).
  4. Hence show that $$\int _ { - 1 } ^ { 3 } \frac { 4 x ^ { 2 } + 8 x } { 2 x + 3 } d x = 12 - 3 \ln 3$$
CAIE P2 2009 June Q8
11 marks Standard +0.3
8
  1. Find the equation of the tangent to the curve \(y = \ln ( 3 x - 2 )\) at the point where \(x = 1\).
    1. Find the value of the constant \(A\) such that $$\frac { 6 x } { 3 x - 2 } \equiv 2 + \frac { A } { 3 x - 2 }$$
    2. Hence show that \(\int _ { 2 } ^ { 6 } \frac { 6 x } { 3 x - 2 } \mathrm {~d} x = 8 + \frac { 8 } { 3 } \ln 2\).
OCR C4 2011 June Q3
8 marks Standard +0.3
3
  1. Find the quotient when \(3 x ^ { 3 } - x ^ { 2 } + 10 x - 3\) is divided by \(x ^ { 2 } + 3\), and show that the remainder is \(x\).
  2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 3 x ^ { 3 } - x ^ { 2 } + 10 x - 3 } { x ^ { 2 } + 3 } \mathrm {~d} x$$