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TMUA 2016 · Paper 2 · Question 1 of 20

TMUA 2016 Paper 2 Question 1

Differentiation and integration — Definite integrals · expanding a bracket before integrating. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Differentiation and integrationDefinite integrals · expanding a bracket before integrating6 options
Find the value of ∫12 (x2 − 4x2)2 dx
  1. A4315
  2. B3
  3. C9715
  4. D10315
  5. E16315
  6. F18
Show the answer and worked solution
answer · A
  1. A4315
  2. B3
  3. C9715
  4. D10315
  5. E16315
  6. F18
There is no shortcut here: square the bracket first so that every term is a power of x. (x2 − 4x−2)2 = x4 − 8 + 16x−4, which integrates to x55 − 8x − 163x3. At x=2 this is 325 − 16 − 23 and at x=1 it is 15 − 8 − 163. Subtracting gives 315 − 8 + 143 = 93 − 120 + 7015 = 4315. The tempting error is to integrate the bracket and then square, which is not a valid operation.