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Year 12 Integration (Polynomials) Worksheet — Level 3 — Practice Set 4 — Answers

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Success TutoringWorksheetsAnswer sheetYear 12 · Mathematics · Calculus

Year 12 Integration (Polynomials) Worksheet — Level 3 — Practice Set 4 — Answers

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

1.Evaluate ∫ from 0 to 2 of (8x3 + 12x2 + 3) dx

Answer: 70

  1. ∫ 8x3 dx = 2x4
  2. ∫ 12x2 dx = 4x3
  3. ∫ 3 dx = 3x
  4. F(x) = 2x4 + 4x3 + 3x
  5. F(2) − F(0) = 70 − (0) = 70
2.Evaluate ∫ from 0 to 1 of (−3x2 − 6x + 3) dx

Answer: −1

  1. ∫ −3x2 dx = −x3
  2. ∫ −6x dx = −3x2
  3. ∫ 3 dx = 3x
  4. F(x) = −x3 − 3x2 + 3x
  5. F(1) − F(0) = −1 − (0) = −1
3.Evaluate ∫ from 2 to 4 of (−8x3 + 4x − 3) dx

Answer: −462

  1. ∫ −8x3 dx = −2x4
  2. ∫ 4x dx = 2x2
  3. ∫ −3 dx = −3x
  4. F(x) = −2x4 + 2x2 − 3x
  5. F(4) − F(2) = −492 − (−30) = −462
4.Evaluate ∫ from 0 to 3 of (−4x3 + 12x2 − 2) dx

Answer: 21

  1. ∫ −4x3 dx = −x4
  2. ∫ 12x2 dx = 4x3
  3. ∫ −2 dx = −2x
  4. F(x) = −x4 + 4x3 − 2x
  5. F(3) − F(0) = 21 − (0) = 21
5.Find ∫ (20x4 + 8x + 3) dx

Answer: 4x5 + 4x2 + 3x + C

  1. ∫ 20x4 dx = 4x5
  2. ∫ 8x dx = 4x2
  3. ∫ 3 dx = 3x
  4. Don't forget the constant of integration, C
6.Evaluate ∫ from 0 to 2 of (−15x4 − 3x2 + 1) dx

Answer: −102

  1. ∫ −15x4 dx = −3x5
  2. ∫ −3x2 dx = −x3
  3. ∫ 1 dx = x
  4. F(x) = −3x5 − x3 + x
  5. F(2) − F(0) = −102 − (0) = −102
7.Evaluate ∫ from 2 to 3 of (−10x4 − 4x3 + 3) dx

Answer: −484

  1. ∫ −10x4 dx = −2x5
  2. ∫ −4x3 dx = −x4
  3. ∫ 3 dx = 3x
  4. F(x) = −2x5 − x4 + 3x
  5. F(3) − F(2) = −558 − (−74) = −484
8.Evaluate ∫ from 2 to 3 of (10x4 + 4x3 − 3x2) dx

Answer: 468

  1. ∫ 10x4 dx = 2x5
  2. ∫ 4x3 dx = x4
  3. ∫ −3x2 dx = −x3
  4. F(x) = 2x5 + x4 − x3
  5. F(3) − F(2) = 540 − (72) = 468
9.Find ∫ (−10x4 − 8x3 − 4x) dx

Answer: −2x5 − 2x4 − 2x2 + C

  1. ∫ −10x4 dx = −2x5
  2. ∫ −8x3 dx = −2x4
  3. ∫ −4x dx = −2x2
  4. Don't forget the constant of integration, C
10.Find ∫ (8x3 − 6x2 + 8x) dx

Answer: 2x4 − 2x3 + 4x2 + C

  1. ∫ 8x3 dx = 2x4
  2. ∫ −6x2 dx = −2x3
  3. ∫ 8x dx = 4x2
  4. Don't forget the constant of integration, C
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