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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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Success TutoringWorksheetsAnswer sheetYear 12 · Mathematics · Calculus

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

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

1.Find ∫ (5x4 + 4x + 1) dx

Answer: x5 + 2x2 + x + C

  1. ∫ 5x4 dx = x5
  2. ∫ 4x dx = 2x2
  3. ∫ 1 dx = x
  4. Don't forget the constant of integration, C
2.Evaluate ∫ from 0 to 2 of (12x3 + 8x − 1) dx

Answer: 62

  1. ∫ 12x3 dx = 3x4
  2. ∫ 8x dx = 4x2
  3. ∫ −1 dx = −x
  4. F(x) = 3x4 + 4x2 − x
  5. F(2) − F(0) = 62 − (0) = 62
3.Find ∫ (5x4 + 3x2 − 3) dx

Answer: x5 + x3 − 3x + C

  1. ∫ 5x4 dx = x5
  2. ∫ 3x2 dx = x3
  3. ∫ −3 dx = −3x
  4. Don't forget the constant of integration, C
4.Evaluate ∫ from 0 to 1 of (6x2 + 2x + 1) dx

Answer: 4

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

Answer: 2x4 − 2x3 + x + C

  1. ∫ 8x3 dx = 2x4
  2. ∫ −6x2 dx = −2x3
  3. ∫ 1 dx = x
  4. Don't forget the constant of integration, C
6.Find ∫ (20x4 + 16x3 − 1) dx

Answer: 4x5 + 4x4 − x + C

  1. ∫ 20x4 dx = 4x5
  2. ∫ 16x3 dx = 4x4
  3. ∫ −1 dx = −x
  4. Don't forget the constant of integration, C
7.Find ∫ (5x4 − 4x − 2) dx

Answer: x5 − 2x2 − 2x + C

  1. ∫ 5x4 dx = x5
  2. ∫ −4x dx = −2x2
  3. ∫ −2 dx = −2x
  4. Don't forget the constant of integration, C
8.Find ∫ (16x3 + 8x − 3) dx

Answer: 4x4 + 4x2 − 3x + C

  1. ∫ 16x3 dx = 4x4
  2. ∫ 8x dx = 4x2
  3. ∫ −3 dx = −3x
  4. Don't forget the constant of integration, C
9.Evaluate ∫ from 2 to 3 of (5x4 + 12x3 − 3) dx

Answer: 403

  1. ∫ 5x4 dx = x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ −3 dx = −3x
  4. F(x) = x5 + 3x4 − 3x
  5. F(3) − F(2) = 477 − (74) = 403
10.Evaluate ∫ from 0 to 3 of (12x3 + 8x + 2) dx

Answer: 285

  1. ∫ 12x3 dx = 3x4
  2. ∫ 8x dx = 4x2
  3. ∫ 2 dx = 2x
  4. F(x) = 3x4 + 4x2 + 2x
  5. F(3) − F(0) = 285 − (0) = 285
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