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.

1 questions

AQA C2 2011 June Q3
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)