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12th Grade Integration (Polynomials) Worksheet — Level 3 — Practice Set 3 — Answers

Answer sheet with worked solutions for the 12th Grade calculus worksheet.

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12th Grade Integration (Polynomials) Worksheet — Level 3 — Practice Set 3 — Answers

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

1.Find ∫ (8x3 − 6x2 − 2) dx

Answer: 2x4 − 2x3 − 2x + C

  1. ∫ 8x3 dx = 2x4
  2. ∫ −6x2 dx = −2x3
  3. ∫ −2 dx = −2x
  4. Don't forget the constant of integration, C
2.Evaluate ∫ from 0 to 2 of (−6x2 − 4x − 2) dx

Answer: −28

  1. ∫ −6x2 dx = −2x3
  2. ∫ −4x dx = −2x2
  3. ∫ −2 dx = −2x
  4. F(x) = −2x3 − 2x2 − 2x
  5. F(2) − F(0) = −28 − (0) = −28
3.Evaluate ∫ from 2 to 4 of (−15x4 + 8x + 2) dx

Answer: −2924

  1. ∫ −15x4 dx = −3x5
  2. ∫ 8x dx = 4x2
  3. ∫ 2 dx = 2x
  4. F(x) = −3x5 + 4x2 + 2x
  5. F(4) − F(2) = −3000 − (−76) = −2924
4.Evaluate ∫ from 2 to 3 of (−8x3 − 3x2 − 4x) dx

Answer: −159

  1. ∫ −8x3 dx = −2x4
  2. ∫ −3x2 dx = −x3
  3. ∫ −4x dx = −2x2
  4. F(x) = −2x4 − x3 − 2x2
  5. F(3) − F(2) = −207 − (−48) = −159
5.Evaluate ∫ from 1 to 2 of (20x4 − 6x2 − 4x) dx

Answer: 104

  1. ∫ 20x4 dx = 4x5
  2. ∫ −6x2 dx = −2x3
  3. ∫ −4x dx = −2x2
  4. F(x) = 4x5 − 2x3 − 2x2
  5. F(2) − F(1) = 104 − (0) = 104
6.Find ∫ (4x3 + 12x2 + 4) dx

Answer: x4 + 4x3 + 4x + C

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

Answer: −9576

  1. ∫ −15x4 dx = −3x5
  2. ∫ −6x2 dx = −2x3
  3. ∫ −6x dx = −3x2
  4. F(x) = −3x5 − 2x3 − 3x2
  5. F(5) − F(2) = −9700 − (−124) = −9576
8.Evaluate ∫ from 2 to 3 of (5x4 − 8x3 − 3x2) dx

Answer: 62

  1. ∫ 5x4 dx = x5
  2. ∫ −8x3 dx = −2x4
  3. ∫ −3x2 dx = −x3
  4. F(x) = x5 − 2x4 − x3
  5. F(3) − F(2) = 54 − (−8) = 62
9.Evaluate ∫ from 1 to 4 of (12x3 − 2x + 4) dx

Answer: 762

  1. ∫ 12x3 dx = 3x4
  2. ∫ −2x dx = −x2
  3. ∫ 4 dx = 4x
  4. F(x) = 3x4 − x2 + 4x
  5. F(4) − F(1) = 768 − (6) = 762
10.Find ∫ (3x2 − 4x − 2) dx

Answer: x3 − 2x2 − 2x + C

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