Mathematics Advanced • Year 11 • Module 3 • Lesson 10

Integration

Build procedural fluency in the power rule for integration, including +C, simplification before integrating, and verification by differentiation.

Build · Skill Drill

1. Quick recall

Answer each in the space provided. 1 mark each

Q1.1 Complete the power rule for integration (n ≠ −1):

∫ xⁿ dx = ______________ + ______

Q1.2 Two facts about the constant of integration:

(a) Every indefinite integral must end with ______.

(b) The +C exists because the derivative of any ____________ is zero, so many functions share the same derivative.

Q1.3 Quick check: you can verify ∫ f(x) dx = F(x) + C by ______________________ F(x) and confirming you obtain f(x).

Stuck? Revisit lesson § Concept (formula box) and § Key Terms.

2. Worked example — ∫ (4x³ − 2x + 5) dx

Integrate term by term, then add +C and verify.

Problem. Find ∫ (4x³ − 2x + 5) dx.

Step 1 — Integrate the cubic term.

∫ 4x³ dx = 4 · x⁴/4 = x⁴

Reason: power rule — raise the index by 1, divide by the new index.

Step 2 — Integrate the linear term.

∫ (−2x) dx = −2 · x²/2 = −x²

Reason: same rule; mind the sign.

Step 3 — Integrate the constant.

∫ 5 dx = 5x

Reason: 5 = 5x⁰, so ∫ 5x⁰ dx = 5x¹/1 = 5x.

Step 4 — Combine and add +C.

Answer: x⁴ − x² + 5x + C

Step 5 — Verify by differentiating.

d/dx (x⁴ − x² + 5x + C) = 4x³ − 2x + 5 ✓

Conclusion. ∫ (4x³ − 2x + 5) dx = x⁴ − x² + 5x + C.

3. Faded example — ∫ (x⁴ + 3x²)/x dx

This one needs simplification before integration. Fill in each blank. 4 marks

Step 1 — Simplify the integrand (split the fraction, divide each term by x):

(x⁴ + 3x²)/x = ______________ + ______________

Step 2 — Integrate term by term:

∫ ______ dx = ______________ ∫ ______ dx = ______________

Step 3 — Combine and add +C:

∫ (x⁴ + 3x²)/x dx = ______________________ + ______

Step 4 — Verify by differentiating:

d/dx [your answer] = ______________ which equals the original integrand ____________________

Stuck? Revisit lesson § Worked Example 3 (simplify first).

4. Graduated practice — find each integral

Include +C in every indefinite integral, simplify the answer where possible.

Foundation — single-term integrals (4 questions)

Q	Integral	Answer (with +C)
4.1 1	∫ 5x⁴ dx
4.2 1	∫ 6x dx
4.3 1	∫ 7 dx
4.4 1	∫ x² dx

Standard — polynomials and "simplify first" (6 questions)

4.5 ∫ (2x + 3) dx 1 mark

4.6 ∫ (x³ − 4x² + x) dx 2 marks

4.7 ∫ (4x³ − 6x² + 2) dx 2 marks

4.8 ∫ (x⁵ − 2x³)/x² dx 2 marks

4.9 ∫ (2x⁴ + 3x²)/x dx 2 marks

4.10 ∫ (x + 2)² dx (hint: expand first) 2 marks

Extension — recover a function from its derivative (2 questions)

4.11 Given f′(x) = 3x² − 2x and f(1) = 4, find f(x). 3 marks

4.12 Given f′(x) = 6x² − 4x + 1 and f(2) = 10, find f(x). 3 marks

Stuck on 4.11 / 4.12? Integrate first to get f(x) + C, then use the given f(value) to solve for C.

5. Self-check the easy 3

Tick the first three once you have checked.

For 4.1 my answer is x⁵ + C (not 5x⁵/5 + C); I simplified the coefficient.

For 4.3 (∫ 7 dx) my answer is 7x + C (not just 7 or 7 + C).

I have +C at the end of every Standard answer. Indefinite integrals without +C are wrong, not just sloppy.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Power rule for integration

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1).

Q1.2 — Constant of integration

(a) Every indefinite integral must end with +C. (b) Because the derivative of any constant is zero, so many functions share the same derivative.

Q1.3 — Verification

Verify by differentiating F(x) and confirming you obtain f(x).

Q3 — Faded example ∫ (x⁴ + 3x²)/x dx

Step 1: (x⁴ + 3x²)/x = x³ + 3x. Step 2: ∫ x³ dx = x⁴/4; ∫ 3x dx = 3x²/2. Step 3: ∫ (x⁴ + 3x²)/x dx = x⁴/4 + 3x²/2 + C. Step 4: d/dx [x⁴/4 + 3x²/2 + C] = x³ + 3x, which equals the simplified integrand x³ + 3x. ✓

Q4.1 — ∫ 5x⁴ dx

= 5 · x⁵/5 + C = x⁵ + C.

Q4.2 — ∫ 6x dx

= 6 · x²/2 + C = 3x² + C.

Q4.3 — ∫ 7 dx

= 7x + C.

Q4.4 — ∫ x² dx

= x³/3 + C.

Q4.5 — ∫ (2x + 3) dx

= x² + 3x + C.

Q4.6 — ∫ (x³ − 4x² + x) dx

= x⁴/4 − 4x³/3 + x²/2 + C.

Q4.7 — ∫ (4x³ − 6x² + 2) dx

∫ 4x³ = x⁴; ∫ −6x² = −2x³; ∫ 2 = 2x. Answer: x⁴ − 2x³ + 2x + C.

Q4.8 — ∫ (x⁵ − 2x³)/x² dx

Simplify first: (x⁵ − 2x³)/x² = x³ − 2x. ∫ (x³ − 2x) dx = x⁴/4 − x² + C.

Q4.9 — ∫ (2x⁴ + 3x²)/x dx

Simplify first: (2x⁴ + 3x²)/x = 2x³ + 3x. ∫ (2x³ + 3x) dx = (2x⁴)/4 + (3x²)/2 + C = x⁴/2 + 3x²/2 + C.

Q4.10 — ∫ (x + 2)² dx

Expand: (x + 2)² = x² + 4x + 4. ∫ = x³/3 + 2x² + 4x + C. Answer: x³/3 + 2x² + 4x + C.

Q4.11 — f′(x) = 3x² − 2x with f(1) = 4

f(x) = ∫ (3x² − 2x) dx = x³ − x² + C. Use f(1) = 4: 1 − 1 + C = 4 ⇒ C = 4. So f(x) = x³ − x² + 4.

Q4.12 — f′(x) = 6x² − 4x + 1 with f(2) = 10

f(x) = ∫ (6x² − 4x + 1) dx = 2x³ − 2x² + x + C. Use f(2) = 10: 16 − 8 + 2 + C = 10 ⇒ 10 + C = 10 ⇒ C = 0. So f(x) = 2x³ − 2x² + x.