Integration toolkit: 7 exam questions solved step by step

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What this lesson covers

  • Integration reverses differentiation, so every standard derivative gives a standard integral.
  • The reverse chain rule divides by the inside derivative, and that factor is the one most often dropped.
Key idea

7 questions, one method

  • 1. Rewrite: surds and fractions as powers.
  • 2. Raise the power, divide by the new one.
  • 3. Reverse chain: ÷ the inside derivative.
  • 4. Differentiate back to check.
Your turn

Question 1 · Achieved

  • Find ∫ (4x³ − 6/x³ + 2) dx
Worked example

Q1: ∫ (4x³ − 6/x³ + 2) dx

  • A fraction: make it a power.
  • Powers rise by one.
  1. ∫ (4x³ − 6/x³ + 2) dx
  2. = ∫ (4x³ − 6x⁻³ + 2) dxas powers
  3. 4x³ → 4x⁴/4 = x⁴power 3 to 4, ÷ 4
  4. −6x⁻³ → −6x⁻²/(−2) = 3x⁻²power −3 to −2
  5. 2 → 2x2 = 2x⁰
  6. = x⁴ + 3x⁻² + 2x + calways + c
  7. = x⁴ + 3/x² + 2x + c
  8. d/dx: 4x³ − 6x⁻³ + 2
Your turn

Question 2 · Achieved

  • Find ∫ (10x√x − 3/√x) dx
Worked example

Q2: ∫ (10x√x − 3/√x) dx

  • Surds as powers first.
  • Add powers when multiplying.
  1. x√x = x¹ × x^(1/2) = x^(3/2)add the powers
  2. x√x = x^(1/2)the x is lost
  3. 3/√x = 3x^(−1/2)
  4. ∫ (10x^(3/2) − 3x^(−1/2)) dx
  5. 10x^(5/2) ÷ (5/2) = 4x^(5/2)power 3/2 to 5/2
  6. −3x^(1/2) ÷ (1/2) = −6x^(1/2)
  7. = 4x^(5/2) − 6√x + c
  8. d/dx: 10x^(3/2) − 3x^(−1/2)
Your turn

Question 3 · Merit

  • Find ∫ (6e^(3x) + 4 sin 2x) dx
Worked example

Q3: ∫ (6e^(3x) + 4 sin 2x) dx

  • Reverse each derivative.
  • Then divide by a.
  1. ∫ e^(ax) dx = (1/a)e^(ax)divide by a
  2. ∫ sin(ax) dx = −(1/a)cos(ax)
  3. 6e^(3x) → 6 × (1/3)e^(3x)÷ 3
  4. = 2e^(3x)
  5. 4 sin 2x → 4 × −(1/2)cos 2xsin to −cos, ÷ 2
  6. = −2cos 2x
  7. = 2e^(3x) − 2cos 2x + c
  8. d/dx: 6e^(3x) + 4 sin 2x
Your turn

Question 4 · Merit

  • Find ∫ 6√(3x + 1) dx
  • Careful: the inside is 3x + 1, not x.
Worked example

Q4: ∫ 6√(3x + 1) dx

  • ÷ the new power.
  • ÷ the inside derivative.
  1. 6√(3x + 1) = 6(3x + 1)^(1/2)
  2. raise: (3x + 1)^(3/2)power 1/2 to 3/2
  3. ÷ 3/2 and ÷ 33 = inside derivative
  4. 6 ÷ (3/2 × 3) = 6 ÷ 9/2 = 4/3
  5. = (4/3)(3x + 1)^(3/2) + c
  6. 4(3x + 1)^(3/2) + cthe ÷ 3 dropped
  7. d/dx gives 18√(3x + 1)three times too big
  8. d/dx: (4/3)(3/2)(3)√(3x + 1)
  9. = 6√(3x + 1)
Your turn

Question 5 · Merit, in context

  • Rain runs into a farm tank at
  • r(t) = 40e^(−0.5t) litres per minute,
  • t minutes after a downpour starts.
  • How much flows in in the first 4 min?
Worked example

Q5: water in the first 4 minutes

  • Amount: integrate the rate.
  • Definite: no + c.
  1. V = ∫₀⁴ 40e^(−0.5t) dtamount = ∫ rate
  2. 40 ÷ (−0.5) = −80divide by a = −0.5
  3. = [−80e^(−0.5t)]₀⁴no + c: definite
  4. = −80e^(−2) − (−80e⁰)top minus bottom
  5. = 80 − 80e^(−2)e⁰ = 1
  6. = 80 − 10.827 = 69.173
  7. about 69.2 L in the first 4 min
  8. 0 < 69.2 < 40 × 4 = 160the rate falls from 40
Your turn

Question 6 · Excellence

  • Find ∫ (sin x + cos x)² dx
Worked example

Q6: ∫ (sin x + cos x)² dx

  • Not a bracket power.
  • Expand, then identities.
  1. (sin x + cos x)³/3inside not linear
  2. (sin x + cos x)²
  3. = sin²x + 2sin x cos x + cos²x
  4. sin²x + cos²x = 1Pythagorean identity
  5. 2sin x cos x = sin 2xdouble angle
  6. ∫ (1 + sin 2x) dx
  7. ∫ 1 dx = x
  8. ∫ sin 2x dx = −(1/2)cos 2xsin to −cos, ÷ 2
  9. = x − (1/2)cos 2x + c
Your turn

Question 7 · Excellence

  • Find ∫ sin 5x sin x dx
  • Use the formula booklet.
Worked example

Q7: ∫ sin 5x sin x dx

  • A product: make it a sum.
  • Then each cos, ÷ a.
  1. 2sin A sin B = cos(A−B) − cos(A+B)
  2. A = 5x, B = x
  3. 2sin 5x sin x = cos 4x − cos 6x
  4. sin 5x sin x = (cos 4x − cos 6x)/2
  5. ∫ (1/2)cos 4x dx = (1/8)sin 4x
  6. ∫ (1/2)cos 6x dx = (1/12)sin 6x
  7. = (1/8)sin 4x − (1/12)sin 6x + c
  8. d/dx: (1/2)cos 4x − (1/2)cos 6x
Summary

The method that solved all seven

  • Rewrite, integrate each term, divide by the inside, check.
StepHow
Surdswrite them as powers
xⁿraise, then divide
e, sin, cosreverse, then × 1/a
(ax + b)ⁿ÷ new power, ÷ a
Identitiesrewrite trig first
Checkdifferentiate back

More in AS 91579

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