Integrating fractions: 7 exam questions solved step by step

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What this lesson covers

  • ln f(x) is the answer only when the numerator is the derivative of the denominator.
  • Other fractions are split, divided, or integrated by reverse chain rule.
Key idea

7 questions, four routes

  • 1. Top = derivative of bottom: ln.
  • 2. One-term bottom: split.
  • 3. Top power ≥ bottom: divide first.
  • 4. Bracket power: reverse chain, not ln.
Your turn

Question 1 · Achieved

  • Find ∫ (2x + 3)/(x² + 3x − 1) dx
Worked example

Q1: (2x + 3)/(x² + 3x − 1)

  • Differentiate the bottom.
  • Does it match the top?
  1. ∫ (2x + 3)/(x² + 3x − 1) dx
  2. d/dx (x² + 3x − 1) = 2x + 3derivative of bottom
  3. top = 2x + 3: a match
  4. = ln|x² + 3x − 1| + cbottom can be < 0
  5. d/dx: (2x + 3)/(x² + 3x − 1)
Your turn

Question 2 · Achieved

  • Find ∫ 4/(3 − 2x) dx
  • Careful with the sign.
Worked example

Q2: ∫ 4/(3 − 2x) dx

  • Bottom′ is −2, not 2.
  • Top as a multiple of it.
  1. ∫ 4/(3 − 2x) dx
  2. d/dx (3 − 2x) = −2derivative of bottom
  3. 4 = −2 × −2top as a multiple
  4. = −2 ∫ −2/(3 − 2x) dx
  5. = −2ln|3 − 2x| + c
  6. 2ln|3 − 2x| + csign lost
  7. d/dx: −2(−2)/(3 − 2x) = 4/(3 − 2x)
Your turn

Question 3 · Merit

  • Find ∫ (x³ − 4x + 6)/x² dx
Worked example

Q3: (x³ − 4x + 6)/x²

  • One-term bottom: split.
  • Only 1/x gives a log.
  1. ∫ (x³ − 4x + 6)/x² dx
  2. ln(x²) + ctop is not 2x
  3. = ∫ (x − 4/x + 6x⁻²) dxeach term ÷ x²
  4. ∫ x dx = x²/2
  5. ∫ −4/x dx = −4ln|x|the only log
  6. ∫ 6x⁻² dx = 6x⁻¹/(−1) = −6/x
  7. = x²/2 − 4ln|x| − 6/x + c
Your turn

Question 4 · Merit

  • Find ∫ (2x² + x − 4)/(x − 1) dx
Worked example

Q4: (2x² + x − 4)/(x − 1)

  • Top power 2, bottom 1.
  • Divide, then integrate.
  1. (2x² + x − 4) ÷ (x − 1)top power ≥ bottom
  2. 2x(x − 1) = 2x² − 2x2x into 2x²
  3. leaves x + 2x − 4 = 3x − 4
  4. 3(x − 1) = 3x − 33 into 3x
  5. leaves −4 + 3 = −1
  6. quotient 2x + 3, remainder −1
  7. (x − 1)(2x + 3) − 1 = 2x² + x − 4
  8. ∫ (2x + 3 − 1/(x − 1)) dx
  9. = x² + 3x − ln|x − 1| + c
Your turn

Question 5 · Merit

  • Find ∫ 6/(2x + 5)⁴ dx
Worked example

Q5: ∫ 6/(2x + 5)⁴ dx

  • A bracket power: not ln.
  • ÷ new power, ÷ inside.
  1. ∫ 6/(2x + 5)⁴ dx
  2. ln|(2x + 5)⁴|derivative 8/(2x + 5)
  3. = ∫ 6(2x + 5)⁻⁴ dxas a power
  4. raise: (2x + 5)⁻³power −4 to −3
  5. 6 ÷ (−3 × 2) = −1÷ new power, ÷ 2
  6. = −(2x + 5)⁻³ + c
  7. = −1/(2x + 5)³ + c
  8. d/dx: −(−3)(2x + 5)⁻⁴ × 2
  9. = 6(2x + 5)⁻⁴
Your turn

Question 6 · Excellence

  • Given ∫₁ᵏ 4/(2x − 1) dx = ln 25
  • and k > 1, find k.
Worked example

Q6: find k

  • Integrate, then the limits.
  • Two roots: test each.
  1. 4 = 2 × 2, so ∫ = 2ln|2x − 1|
  2. [2ln|2x − 1|]₁ᵏ = ln 25no + c: definite
  3. 2ln(2k − 1) − 2ln 1 = ln 25k > 1: 2k − 1 > 0
  4. 2ln(2k − 1) = ln 25ln 1 = 0
  5. ln (2k − 1)² = ln 252ln a = ln a²
  6. (2k − 1)² = 25equal logs
  7. 2k − 1 = 5 or 2k − 1 = −5
  8. k = −2k must be > 1
  9. k = 3
Your turn

Question 7 · Excellence, in context

  • Water is pumped into a trough at
  • r(t) = (t² + 3t + 4)/(t + 1) L/min.
  • How much water goes in
  • during the first 3 minutes?
Worked example

Q7: divide first

  • Amount: integrate the rate.
  • Top power ≥ bottom.
  1. V = ∫₀³ (t² + 3t + 4)/(t + 1) dt
  2. top power 2 ≥ bottom power 1
  3. t(t + 1) = t² + tt into t²
  4. leaves 3t − t + 4 = 2t + 4
  5. 2(t + 1) = 2t + 22 into 2t
  6. leaves 4 − 2 = 2
  7. quotient t + 2, remainder 2
  8. (t + 1)(t + 2) + 2 = t² + 3t + 4
Worked example

Q7: integrate over 0 to 3

  • Definite: no + c.
  • Answer in litres.
  1. V = ∫₀³ (t + 2 + 2/(t + 1)) dt
  2. = [t²/2 + 2t + 2ln(t + 1)]₀³
  3. t + 1 > 0, so no | | needed
  4. t = 3: 9/2 + 6 + 2ln 4top limit
  5. t = 0: 0 + 0 + 2ln 1 = 0bottom limit
  6. V = 10.5 + 2ln 4
  7. = 10.5 + 2.773 = 13.273
  8. about 13.3 L in the first 3 min
Summary

The routes that solved all seven

  • Check the top against the bottom before you write ln.
StepHow
ln casetop is bottom′
Splitone-term bottom
Dividetop power ≥ bottom
Powerbracket: reverse chain
Limitsno + c, reject roots
Checkdifferentiate back

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