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Year 12 Integration (Polynomials) Worksheet — Level 3 — Practice Set 2 — Answers

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Success TutoringWorksheetsAnswer sheetYear 12 · Mathematics · Calculus

Year 12 Integration (Polynomials) Worksheet — Level 3 — Practice Set 2 — Answers

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

1.Evaluate ∫ from 2 to 3 of (15x4 − 8x3 + 4x) dx

Answer: 513

  1. ∫ 15x4 dx = 3x5
  2. ∫ −8x3 dx = −2x4
  3. ∫ 4x dx = 2x2
  4. F(x) = 3x5 − 2x4 + 2x2
  5. F(3) − F(2) = 585 − (72) = 513
2.Find ∫ (−6x2 + 6x + 2) dx

Answer: −2x3 + 3x2 + 2x + C

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

Answer: −42

  1. ∫ −4x3 dx = −x4
  2. ∫ 6x2 dx = 2x3
  3. ∫ −6x dx = −3x2
  4. F(x) = −x4 + 2x3 − 3x2
  5. F(3) − F(2) = −54 − (−12) = −42
4.Find ∫ (−5x4 + 12x3 + 1) dx

Answer: −x5 + 3x4 + x + C

  1. ∫ −5x4 dx = −x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ 1 dx = x
  4. Don't forget the constant of integration, C
5.Evaluate ∫ from 2 to 5 of (−10x4 − 12x3 + 4x) dx

Answer: −7971

  1. ∫ −10x4 dx = −2x5
  2. ∫ −12x3 dx = −3x4
  3. ∫ 4x dx = 2x2
  4. F(x) = −2x5 − 3x4 + 2x2
  5. F(5) − F(2) = −8075 − (−104) = −7971
6.Find ∫ (12x3 + 4x − 1) dx

Answer: 3x4 + 2x2 − x + C

  1. ∫ 12x3 dx = 3x4
  2. ∫ 4x dx = 2x2
  3. ∫ −1 dx = −x
  4. Don't forget the constant of integration, C
7.Evaluate ∫ from 2 to 5 of (10x4 + 6x2 − 6x) dx

Answer: 6357

  1. ∫ 10x4 dx = 2x5
  2. ∫ 6x2 dx = 2x3
  3. ∫ −6x dx = −3x2
  4. F(x) = 2x5 + 2x3 − 3x2
  5. F(5) − F(2) = 6425 − (68) = 6357
8.Find ∫ (15x4 + 6x + 4) dx

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

  1. ∫ 15x4 dx = 3x5
  2. ∫ 6x dx = 3x2
  3. ∫ 4 dx = 4x
  4. Don't forget the constant of integration, C
9.Evaluate ∫ from 2 to 3 of (10x4 − 6x2 + 2) dx

Answer: 386

  1. ∫ 10x4 dx = 2x5
  2. ∫ −6x2 dx = −2x3
  3. ∫ 2 dx = 2x
  4. F(x) = 2x5 − 2x3 + 2x
  5. F(3) − F(2) = 438 − (52) = 386
10.Evaluate ∫ from 0 to 2 of (12x3 + 8x + 4) dx

Answer: 72

  1. ∫ 12x3 dx = 3x4
  2. ∫ 8x dx = 4x2
  3. ∫ 4 dx = 4x
  4. F(x) = 3x4 + 4x2 + 4x
  5. F(2) − F(0) = 72 − (0) = 72
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