Direct binomial expansion then integrate

Questions that ask to expand a single binomial expression and then integrate it directly, without any algebraic manipulation or combination of multiple expansions.

6 questions · Moderate -0.8

1.04a Binomial expansion: (a+b)^n for positive integer n1.08b Integrate x^n: where n != -1 and sums
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OCR C2 2005 June Q6
8 marks Moderate -0.8
6
  1. Find the binomial expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 } \mathrm {~d} x\).
OCR C2 2009 June Q4
8 marks Moderate -0.8
4
  1. Find the binomial expansion of \(\left( x ^ { 2 } - 5 \right) ^ { 3 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 2 } - 5 \right) ^ { 3 } \mathrm {~d} x\).
OCR C2 2014 June Q6
9 marks Moderate -0.8
6
  1. Find the binomial expansion of \(\left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 3 } + \frac { 2 } { x ^ { 2 } } \right) ^ { 4 } \mathrm {~d} x\).
AQA C2 2008 June Q7
9 marks Moderate -0.8
7
  1. The expression \(\left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 }\) can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } + \frac { 64 } { x ^ { 6 } }$$ By using the binomial expansion, or otherwise, find the values of the integers \(p\) and \(q\).
    1. Hence find \(\int \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
    2. Hence find the value of \(\int _ { 1 } ^ { 2 } \left( 1 + \frac { 4 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
AQA C2 2010 June Q4
8 marks Moderate -0.8
4
  1. The expression \(\left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 }\) can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } - \frac { 1 } { x ^ { 6 } }$$ Find the values of the integers \(p\) and \(q\).
    1. Hence find \(\int \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
    2. Hence find the value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
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OCR C2 Q6
8 marks Moderate -0.8
  1. Find the binomial expansion of \(\left(x^2 + \frac{1}{x}\right)^3\), simplifying the terms. [4]
  2. Hence find \(\int \left(x^2 + \frac{1}{x}\right)^3 dx\). [4]