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Year 12 Integration (Polynomials) Worksheet — Level 3 — Answers

Answer sheet with worked solutions for the Year 12 calculus worksheet.

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Year 12 · Mathematics · Calculus

Year 12 Integration (Polynomials) Worksheet — Level 3 — Answers

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Use the rule ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C.

1.Evaluate ∫ from 1 to 2 of (5x4 + 8x3 − 2) dx

Answer: 59

  1. ∫ 5x4 dx = x5
  2. ∫ 8x3 dx = 2x4
  3. ∫ −2 dx = −2x
  4. F(x) = x5 + 2x4 − 2x
  5. F(2) − F(1) = 60 − (1) = 59
2.Find ∫ (−4x3 − 3x2 + 2) dx

Answer: −x4 − x3 + 2x + C

  1. ∫ −4x3 dx = −x4
  2. ∫ −3x2 dx = −x3
  3. ∫ 2 dx = 2x
  4. Don't forget the constant of integration, C
3.Evaluate ∫ from 0 to 2 of (20x4 − 4x3 − 1) dx

Answer: 110

  1. ∫ 20x4 dx = 4x5
  2. ∫ −4x3 dx = −x4
  3. ∫ −1 dx = −x
  4. F(x) = 4x5 − x4 − x
  5. F(2) − F(0) = 110 − (0) = 110
4.Find ∫ (−6x2 + 8x − 3) dx

Answer: −2x3 + 4x2 − 3x + C

  1. ∫ −6x2 dx = −2x3
  2. ∫ 8x dx = 4x2
  3. ∫ −3 dx = −3x
  4. Don't forget the constant of integration, C
5.Find ∫ (−9x2 − 2x − 3) dx

Answer: −3x3 − x2 − 3x + C

  1. ∫ −9x2 dx = −3x3
  2. ∫ −2x dx = −x2
  3. ∫ −3 dx = −3x
  4. Don't forget the constant of integration, C
6.Evaluate ∫ from 1 to 3 of (−15x4 − 8x3 − 1) dx

Answer: −888

  1. ∫ −15x4 dx = −3x5
  2. ∫ −8x3 dx = −2x4
  3. ∫ −1 dx = −x
  4. F(x) = −3x5 − 2x4 − x
  5. F(3) − F(1) = −894 − (−6) = −888
7.Evaluate ∫ from 2 to 3 of (−10x4 + 8x + 3) dx

Answer: −399

  1. ∫ −10x4 dx = −2x5
  2. ∫ 8x dx = 4x2
  3. ∫ 3 dx = 3x
  4. F(x) = −2x5 + 4x2 + 3x
  5. F(3) − F(2) = −441 − (−42) = −399
8.Evaluate ∫ from 0 to 2 of (10x4 + 12x3 + 1) dx

Answer: 114

  1. ∫ 10x4 dx = 2x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ 1 dx = x
  4. F(x) = 2x5 + 3x4 + x
  5. F(2) − F(0) = 114 − (0) = 114
9.Evaluate ∫ from 0 to 3 of (−15x4 + 16x3 − 1) dx

Answer: −408

  1. ∫ −15x4 dx = −3x5
  2. ∫ 16x3 dx = 4x4
  3. ∫ −1 dx = −x
  4. F(x) = −3x5 + 4x4 − x
  5. F(3) − F(0) = −408 − (0) = −408
10.Find ∫ (10x4 + 12x3 + 12x2) dx

Answer: 2x5 + 3x4 + 4x3 + C

  1. ∫ 10x4 dx = 2x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ 12x2 dx = 4x3
  4. Don't forget the constant of integration, C
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