Success TutoringWorksheets

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

Answer sheet with worked solutions for the Year 12 calculus worksheet.

Download answers PDFPrint answersBack to worksheet
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.Find ∫ (10x4 − 2x + 1) dx

Answer: 2x5 − x2 + x + C

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

Answer: −35

  1. ∫ −5x4 dx = −x5
  2. ∫ −4x dx = −2x2
  3. ∫ 2 dx = 2x
  4. F(x) = −x5 − 2x2 + 2x
  5. F(2) − F(1) = −36 − (−1) = −35
3.Evaluate ∫ from 0 to 3 of (−10x4 − 9x2 − 6x) dx

Answer: −594

  1. ∫ −10x4 dx = −2x5
  2. ∫ −9x2 dx = −3x3
  3. ∫ −6x dx = −3x2
  4. F(x) = −2x5 − 3x3 − 3x2
  5. F(3) − F(0) = −594 − (0) = −594
4.Find ∫ (12x3 + 4x + 4) dx

Answer: 3x4 + 2x2 + 4x + C

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

Answer: 32

  1. ∫ 6x2 dx = 2x3
  2. ∫ −4x dx = −2x2
  3. ∫ 4 dx = 4x
  4. F(x) = 2x3 − 2x2 + 4x
  5. F(3) − F(2) = 48 − (16) = 32
6.Evaluate ∫ from 0 to 1 of (−5x4 − 8x3 − 2) dx

Answer: −5

  1. ∫ −5x4 dx = −x5
  2. ∫ −8x3 dx = −2x4
  3. ∫ −2 dx = −2x
  4. F(x) = −x5 − 2x4 − 2x
  5. F(1) − F(0) = −5 − (0) = −5
7.Evaluate ∫ from 0 to 3 of (20x4 + 8x3 + 3) dx

Answer: 1143

  1. ∫ 20x4 dx = 4x5
  2. ∫ 8x3 dx = 2x4
  3. ∫ 3 dx = 3x
  4. F(x) = 4x5 + 2x4 + 3x
  5. F(3) − F(0) = 1143 − (0) = 1143
8.Find ∫ (−10x4 − 8x3 − 2x) dx

Answer: −2x5 − 2x4 − x2 + C

  1. ∫ −10x4 dx = −2x5
  2. ∫ −8x3 dx = −2x4
  3. ∫ −2x dx = −x2
  4. Don't forget the constant of integration, C
9.Find ∫ (20x4 + 6x + 2) dx

Answer: 4x5 + 3x2 + 2x + C

  1. ∫ 20x4 dx = 4x5
  2. ∫ 6x dx = 3x2
  3. ∫ 2 dx = 2x
  4. Don't forget the constant of integration, C
10.Find ∫ (−5x4 − 4x3 + 9x2) dx

Answer: −x5 − x4 + 3x3 + C

  1. ∫ −5x4 dx = −x5
  2. ∫ −4x3 dx = −x4
  3. ∫ 9x2 dx = 3x3
  4. Don't forget the constant of integration, C
Free worksheets at worksheets.successtutoring.com · © Success Tutoring · Free for personal and classroom use

Marking tips

  • Give credit for correct method even when the final answer slips — the working shows where the thinking went right.
  • For open-ended tasks, use the success criteria as a checklist rather than looking for one "right" answer.
  • Celebrate what went well first, then pick one thing to work on next.

Great work. Here's what to try next: