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

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

1.Evaluate ∫ from 0 to 2 of (10x4 + 8x3 + 4) dx

Answer: 104

  1. ∫ 10x4 dx = 2x5
  2. ∫ 8x3 dx = 2x4
  3. ∫ 4 dx = 4x
  4. F(x) = 2x5 + 2x4 + 4x
  5. F(2) − F(0) = 104 − (0) = 104
2.Evaluate ∫ from 1 to 4 of (8x3 + 6x2 + 8x) dx

Answer: 696

  1. ∫ 8x3 dx = 2x4
  2. ∫ 6x2 dx = 2x3
  3. ∫ 8x dx = 4x2
  4. F(x) = 2x4 + 2x3 + 4x2
  5. F(4) − F(1) = 704 − (8) = 696
3.Evaluate ∫ from 2 to 5 of (−15x4 − 12x3 + 1) dx

Answer: −11103

  1. ∫ −15x4 dx = −3x5
  2. ∫ −12x3 dx = −3x4
  3. ∫ 1 dx = x
  4. F(x) = −3x5 − 3x4 + x
  5. F(5) − F(2) = −11245 − (−142) = −11103
4.Evaluate ∫ from 2 to 3 of (−9x2 − 2x + 2) dx

Answer: −60

  1. ∫ −9x2 dx = −3x3
  2. ∫ −2x dx = −x2
  3. ∫ 2 dx = 2x
  4. F(x) = −3x3 − x2 + 2x
  5. F(3) − F(2) = −84 − (−24) = −60
5.Find ∫ (8x3 − 2x + 3) dx

Answer: 2x4 − x2 + 3x + C

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

Answer: x5 − 3x4 + 4x + C

  1. ∫ 5x4 dx = x5
  2. ∫ −12x3 dx = −3x4
  3. ∫ 4 dx = 4x
  4. Don't forget the constant of integration, C
7.Find ∫ (16x3 + 12x2 + 2x) dx

Answer: 4x4 + 4x3 + x2 + C

  1. ∫ 16x3 dx = 4x4
  2. ∫ 12x2 dx = 4x3
  3. ∫ 2x dx = x2
  4. Don't forget the constant of integration, C
8.Find ∫ (−10x4 − 3x2 + 2x) dx

Answer: −2x5 − x3 + x2 + C

  1. ∫ −10x4 dx = −2x5
  2. ∫ −3x2 dx = −x3
  3. ∫ 2x dx = x2
  4. Don't forget the constant of integration, C
9.Evaluate ∫ from 0 to 2 of (5x4 + 12x3 − 1) dx

Answer: 78

  1. ∫ 5x4 dx = x5
  2. ∫ 12x3 dx = 3x4
  3. ∫ −1 dx = −x
  4. F(x) = x5 + 3x4 − x
  5. F(2) − F(0) = 78 − (0) = 78
10.Find ∫ (5x4 − 4x3 + 4x) dx

Answer: x5 − x4 + 2x2 + C

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