Expansion with algebraic manipulation then integrate

Questions that expand a binomial expression but require dividing by a power of x or other algebraic manipulation before integration.

3 questions · Moderate -0.3

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AQA C2 2011 June Q3
10 marks Standard +0.3
3
  1. The expression \(\left( 2 + x ^ { 2 } \right) ^ { 3 }\) can be written in the form $$8 + p x ^ { 2 } + q x ^ { 4 } + x ^ { 6 }$$ Show that \(p = 12\) and find the value of the integer \(q\).
    1. Hence find \(\int \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (5 marks)
    2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (2 marks)
WJEC Unit 1 2022 June Q13
4 marks Moderate -0.3
Find the term which is independent of \(x\) in the expansion of \(\frac{(2-3x)^5}{x^3}\). [4]
SPS SPS SM Pure 2022 June Q1
6 marks Moderate -0.8
  1. The expression \((2 + x^2)^3\) can be written in the form $$8 + px^2 + qx^4 + x^6$$ Demonstrate clearly, using the binomial expansion, that \(p = 12\) and find the value of \(q\). [3 marks]
  2. Hence find \(\int \frac{(2 + x^2)^3}{x^4} dx\). [3 marks]