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Year 13 Integration (Polynomials) Worksheet — Level 3 — Answers

Answer sheet with worked solutions for the Year 13 calculus worksheet.

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Year 13 · Mathematics · Calculus

Year 13 Integration (Polynomials) Worksheet — Level 3 — Answers

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

1.Evaluate ∫ from 2 to 4 of (4x3 + 9x2 − 6x) dx

Answer: 372

  1. ∫ 4x3 dx = x4
  2. ∫ 9x2 dx = 3x3
  3. ∫ −6x dx = −3x2
  4. F(x) = x4 + 3x3 − 3x2
  5. F(4) − F(2) = 400 − (28) = 372
2.Evaluate ∫ from 2 to 3 of (10x4 + 12x3 − 1) dx

Answer: 616

  1. ∫ 10x4 dx = 2x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ −1 dx = −x
  4. F(x) = 2x5 + 3x4 − x
  5. F(3) − F(2) = 726 − (110) = 616
3.Find ∫ (12x3 − 2x − 1) dx

Answer: 3x4 − x2 − x + C

  1. ∫ 12x3 dx = 3x4
  2. ∫ −2x dx = −x2
  3. ∫ −1 dx = −x
  4. Don't forget the constant of integration, C
4.Evaluate ∫ from 0 to 1 of (10x4 + 6x2 − 2x) dx

Answer: 3

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

Answer: −1251

  1. ∫ −10x4 dx = −2x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ 4x dx = 2x2
  4. F(x) = −2x5 + 3x4 + 2x2
  5. F(4) − F(1) = −1248 − (3) = −1251
6.Find ∫ (10x4 + 6x2 − 4x) dx

Answer: 2x5 + 2x3 − 2x2 + C

  1. ∫ 10x4 dx = 2x5
  2. ∫ 6x2 dx = 2x3
  3. ∫ −4x dx = −2x2
  4. Don't forget the constant of integration, C
7.Find ∫ (−15x4 + 12x2 − 1) dx

Answer: −3x5 + 4x3 − x + C

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

Answer: −84

  1. ∫ −3x2 dx = −x3
  2. ∫ −2x dx = −x2
  3. ∫ −2 dx = −2x
  4. F(x) = −x3 − x2 − 2x
  5. F(4) − F(1) = −88 − (−4) = −84
9.Find ∫ (12x3 − 9x2 − 4x) dx

Answer: 3x4 − 3x3 − 2x2 + C

  1. ∫ 12x3 dx = 3x4
  2. ∫ −9x2 dx = −3x3
  3. ∫ −4x dx = −2x2
  4. Don't forget the constant of integration, C
10.Find ∫ (−10x4 − 6x2 + 2x) dx

Answer: −2x5 − 2x3 + x2 + C

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