NCEA Level 3The antiderivative toolkitAS 91579

The integration toolkit: surds, e, trig and the chain rule

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

Integration undoes differentiation

  • Every derivative, read backwards.
  • The booklet lists the standard ones.
  • Get the question into that form, then + c.
Worked example

∫ (6x² − 4x + 5) dx

  • Add one to the power.
  • Divide by the new power.
  • Always + c.
  1. ∫ (6x² − 4x + 5) dx
  2. 6x² → 6x³/3 = 2x³power 2 to 3, ÷ 3
  3. −4x → −4x²/2 = −2x²power 1 to 2, ÷ 2
  4. 5 → 5x5 = 5x⁰
  5. = 2x³ − 2x² + 5x + calways + c
Worked example

Surds as powers: a 2025 error

  • √x = x^(1/2).
  • 1/√x = x^(−1/2).
  1. √x = x^(1/2), 1/√x = x^(−1/2)
  2. ∫ 3√x dx = ∫ 3x^(1/2) dx
  3. = 3x^(3/2) ÷ (3/2)power 1/2 to 3/2
  4. = 2x^(3/2) + c
  5. ∫ 2/√x dx = ∫ 2x^(−1/2) dx
  6. = 2x^(1/2) ÷ (1/2)power −1/2 to 1/2
  7. = 4x^(1/2) + c = 4√x + c
Worked example

∫ (x − 4)/√x dx: split first

  • Split the fraction first.
  • Then each term is a power.
  1. (x − 4)/√x = x/√x − 4/√xsplit the fraction
  2. = x^(1/2) − 4x^(−1/2)as powers
  3. ∫: x^(3/2)/(3/2) − 4x^(1/2)/(1/2)
  4. = (2/3)x^(3/2) − 8x^(1/2) + c
Worked example

e to the ax + b

  • ∫ eˣ dx = eˣ + c.
  • e^(ax + b): divide by a.
  1. ∫ eˣ dx = eˣ + c
  2. ∫ e^(3x − 1) dx
  3. = (1/3)e^(3x − 1) + c÷ 3
  4. ∫ 4e^(−2x) dx
  5. = 4 × (−1/2)e^(−2x)÷ −2
  6. = −2e^(−2x) + c
  7. ∫ e^(kx) dx = (1/k)e^(kx) + c
Worked example

Sine, cos and sec²

  • sin → −cos, cos → sin.
  • sec² → tan. Radians only.
  • Divide by the x coefficient.
  1. ∫ sin 4x dx
  2. = −(1/4)cos 4x + csin to −cos, ÷ 4
  3. ∫ cos 2x dx
  4. = (1/2)sin 2x + ccos to sin, ÷ 2
  5. ∫ sec² 3x dx
  6. = (1/3)tan 3x + csec² to tan, ÷ 3
Worked example

Reverse chain rule: ∫ (2x + 1)⁵ dx

  • Raise the power by one.
  • Divide by the new power.
  • Divide by inside derivative.
  1. ∫ (2x + 1)⁵ dx
  2. (2x + 1)⁶raise the power to 6
  3. (2x + 1)⁶/6÷ the new power
  4. (2x + 1)⁶/(6 × 2)÷ 2, inside derivative
  5. = (2x + 1)⁶/12 + c
Worked example

The dropped factor: a 2025 error

  • A dropped factor scored 0.
  • Differentiate to check.
  1. d/dx (2x + 1)⁶/6factor dropped
  2. = 6(2x + 1)⁵ × 2/6chain rule
  3. = 2(2x + 1)⁵double: wrong
  4. d/dx (2x + 1)⁶/12
  5. = 6(2x + 1)⁵ × 2/12chain rule
  6. = (2x + 1)⁵
  7. d/dx e^(3x − 1)no 1/3
  8. = 3e^(3x − 1)triple: wrong
Worked example

∫ 1/√(4x + 1) dx

  • Write it as a power.
  • Then reverse chain rule.
  1. 1/√(4x + 1) = (4x + 1)^(−1/2)
  2. raise: (4x + 1)^(1/2)power −1/2 to 1/2
  3. ÷ 1/2: 2(4x + 1)^(1/2)doubles it
  4. ÷ 4: (2/4)(4x + 1)^(1/2)inside derivative
  5. = (1/2)√(4x + 1) + c
Worked example

A trig product becomes a sum

  • Booklet: product to sum.
  • A = 3x, B = x.
  • Integrate each sine.
  1. 2sin A cos B = sin(A+B) + sin(A−B)
  2. A = 3x, B = x
  3. 2sin 3x cos x = sin 4x + sin 2x
  4. ∫ sin 4x dx = −(1/4)cos 4x÷ 4
  5. ∫ sin 2x dx = −(1/2)cos 2x÷ 2
  6. = −(1/4)cos 4x − (1/2)cos 2x + c
Worked example

∫ sin²x dx: rewrite it first

  • Booklet: double angle.
  • Rearrange for sin²x.
  1. cos 2x = 1 − 2sin²xbooklet
  2. 2sin²x = 1 − cos 2xrearrange
  3. sin²x = (1 − cos 2x)/2÷ 2
  4. = 1/2 − (1/2)cos 2xsplit
  5. ∫: x/2 − (1/2)(1/2)sin 2xintegrate each term
  6. = x/2 − (1/4)sin 2x + c
Key idea

Show the integrated function

  • No credit for a calculator answer alone.
  • Unknown constants stop the calculator.
  • Show the rewrite and the identity.
Key idea

Where the marks are

  • Achieved: surds, e, trig, (ax + b)ⁿ.
  • Merit: a trig product into a sum.
  • Excellence: choose the right identity.
Summary

What to take away

  • Rewrite into a booklet form, divide by the inside derivative, add + c.
Surdspowers first
xⁿraise, then divide
e^(ax + b)× 1/a
sin, cos, sec²× 1/a
Products, sin²booklet identity
Checkdifferentiate back
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Question 1: rewrite the surd.
  • Question 4: use an identity.
GivenFind
1∫ (x³ − 2√x) dxthe integral
2∫ 6e^(2x + 1) dxthe integral
3∫ (3x − 2)⁴ dxthe integral
4∫ tan²x dxuse an identity
Show answer

1. x⁴/4 − (4/3)x^(3/2) + c. 2. 3e^(2x + 1) + c. 3. (3x − 2)⁵/15 + c. 4. tan²x = sec²x − 1, so tan x − x + c.

Worked example

The answers: questions 1 and 2

  1. 1. x³ − 2√x = x³ − 2x^(1/2)surd as a power
  2. ∫: x⁴/4 − 2x^(3/2)/(3/2)raise, divide
  3. = x⁴/4 − (4/3)x^(3/2) + c
  4. 2. ∫ 6e^(2x + 1) dx
  5. = 6 × (1/2)e^(2x + 1)÷ 2
  6. = 3e^(2x + 1) + c
Worked example

The answers: questions 3 and 4

  1. 3. ∫ (3x − 2)⁴ dx
  2. (3x − 2)⁵raise the power to 5
  3. (3x − 2)⁵/(5 × 3)÷ 5, then ÷ 3
  4. = (3x − 2)⁵/15 + c
  5. 4. tan²x + 1 = sec²xsin² + cos² = 1, ÷ cos²
  6. tan²x = sec²x − 1
  7. ∫ (sec²x − 1) dx
  8. = tan x − x + c

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