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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 ∫ (−15x4 + 12x3 − 6x) dx

Answer: −3x5 + 3x4 − 3x2 + C

  1. ∫ −15x4 dx = −3x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ −6x dx = −3x2
  4. Don't forget the constant of integration, C
2.Evaluate ∫ from 0 to 1 of (−5x4 + 2x + 3) dx

Answer: 3

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

Answer: 8

  1. ∫ 12x3 dx = 3x4
  2. ∫ 3x2 dx = x3
  3. ∫ 8x dx = 4x2
  4. F(x) = 3x4 + x3 + 4x2
  5. F(1) − F(0) = 8 − (0) = 8
4.Find ∫ (−4x3 + 6x2 − 2x) dx

Answer: −x4 + 2x3 − x2 + C

  1. ∫ −4x3 dx = −x4
  2. ∫ 6x2 dx = 2x3
  3. ∫ −2x dx = −x2
  4. Don't forget the constant of integration, C
5.Evaluate ∫ from 2 to 5 of (15x4 + 4x − 2) dx

Answer: 9315

  1. ∫ 15x4 dx = 3x5
  2. ∫ 4x dx = 2x2
  3. ∫ −2 dx = −2x
  4. F(x) = 3x5 + 2x2 − 2x
  5. F(5) − F(2) = 9415 − (100) = 9315
6.Find ∫ (3x2 + 8x + 1) dx

Answer: x3 + 4x2 + x + C

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

Answer: 4290

  1. ∫ 5x4 dx = x5
  2. ∫ 8x3 dx = 2x4
  3. ∫ −2x dx = −x2
  4. F(x) = x5 + 2x4 − x2
  5. F(5) − F(2) = 4350 − (60) = 4290
8.Evaluate ∫ from 2 to 4 of (20x4 + 8x + 3) dx

Answer: 4022

  1. ∫ 20x4 dx = 4x5
  2. ∫ 8x dx = 4x2
  3. ∫ 3 dx = 3x
  4. F(x) = 4x5 + 4x2 + 3x
  5. F(4) − F(2) = 4172 − (150) = 4022
9.Evaluate ∫ from 2 to 5 of (5x4 − 9x2 − 3) dx

Answer: 2733

  1. ∫ 5x4 dx = x5
  2. ∫ −9x2 dx = −3x3
  3. ∫ −3 dx = −3x
  4. F(x) = x5 − 3x3 − 3x
  5. F(5) − F(2) = 2735 − (2) = 2733
10.Find ∫ (15x4 − 9x2 + 8x) dx

Answer: 3x5 − 3x3 + 4x2 + C

  1. ∫ 15x4 dx = 3x5
  2. ∫ −9x2 dx = −3x3
  3. ∫ 8x dx = 4x2
  4. Don't forget the constant of integration, C
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