Cambridge International AS & A Level Mathematics 9709: Pure Mathematics 2 Integration – Revision Notes

For full explanations and longer worked examples, use the integration study guide.

These notes cover section 2.5, Integration, of the Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2026 and 2027 (Version 4). Section 2.5 belongs to Pure Mathematics 2 and is examined on Paper 2, the AS Level Pure Mathematics route (Paper 1 plus Paper 2). Paper 1 knowledge is assumed, so areas, volumes and (ax + b)ⁿ integrals can be built into Paper 2 questions. A scientific calculator is allowed, but unsupported calculator answers earn no marks.

Links: integration practice questions, Pure Mathematics 1 integration notes, Pure Mathematics 2 overview notes, A Level Mathematics hub, printable 9709 checklist and the free AS diagnostic.

2.5 at a glance

Outcome	In one line
Reverse differentiation	Integrate e^(ax + b), 1/(ax + b), sin(ax + b), cos(ax + b), sec²(ax + b)
Trig relationships	Use identities (double-angle, sec² = 1 + tan²) to reach standard forms
Trapezium rule	Estimate a definite integral; say whether it over- or under-estimates, using a sketch

Not on Paper 2: the general method of substitution (the syllabus says it is not required), integration by parts, partial fractions and the kf′(x)/f(x) pattern. Those are section 3.5, Paper 3.

Standard results

f(x)	∫ f(x) dx	Check by differentiating
e^(ax + b)	(1/a) e^(ax + b) + c	(1/a) × a e^(ax + b)
1/(ax + b)	(1/a) ln|ax + b| + c	(1/a) × a/(ax + b)
sin(ax + b)	−(1/a) cos(ax + b) + c	−(1/a) × (−a sin(ax + b))
cos(ax + b)	(1/a) sin(ax + b) + c	(1/a) × a cos(ax + b)
sec²(ax + b)	(1/a) tan(ax + b) + c	(1/a) × a sec²(ax + b)

Radians only. MF19 prints the a = 1, b = 0 versions; the 1/a is yours to supply.

Method: integrating a linear composite

1. Spot the inside function ax + b and write down a (with its sign).
2. Write the outside result from the table.
3. Multiply by 1/a (and by any constant already in front).
4. Add + c for an indefinite integral.
5. Differentiate back in your head to check the sign and the factor.

Small worked reminders

∫ 12e^(−3x) dx      = 12 × (1/(−3)) e^(−3x)  = −4e^(−3x) + c

∫₀² 3/(4 − x) dx    = [−3 ln|4 − x|]₀²
                    = −3 ln 2 + 3 ln 4      = 3 ln 2

In the second one a = −1, so the minus sign comes out in front. With a negative a the log term is subtracted, so take care over which limit gives which sign.

Trigonometrical relationships

Integrand	Rewrite as	Integral
sin²x	½(1 − cos 2x)	½x − ¼ sin 2x + c
cos²x	½(1 + cos 2x)	½x + ¼ sin 2x + c
sin x cos x	½ sin 2x	−¼ cos 2x + c
tan²x	sec²x − 1	tan x − x + c
cos²(kx)	½(1 + cos 2kx)	½x + (1/(4k)) sin 2kx + c

Memory check for the signs: sin²x is small near x = 0, and ½(1 − cos 2x) = 0 at x = 0. So sin² goes with the minus.

Method: trig integrand that is not in the table

1. Is it a power or product of trig functions?   → identity needed
2. Squares of sin or cos                          → double-angle formula
3. sin × cos of the same angle                    → sin 2A = 2 sin A cos A
4. tan²                                           → sec² − 1
5. Expanded bracket                               → look for sin² + cos² = 1
6. Integrate term by term, then apply limits in radians

Small worked reminders

∫ cos²(½x) dx       = ∫ ½(1 + cos x) dx     = ½x + ½ sin x + c

∫ 2 sin 3x cos 3x dx = ∫ sin 6x dx          = −(1/6) cos 6x + c

In the first, the angle doubles from ½x to x. In the second, 2 sin A cos A with A = 3x is sin 6x.

The trapezium rule

∫ₐᵇ y dx ≈ ½h { y₀ + 2(y₁ + … + yₙ₋₁) + yₙ },    h = (b − a)/n

Not printed in MF19. n strips, n + 1 ordinates.

Method: trapezium rule questions

1. h = (b − a)/n
2. Table of x and y (4 d.p. or better; radians for trig)
3. Ends once, middles twice, times ½h
4. Round only at the end (3 s.f. unless told otherwise)
5. Over/under: sketch, look at which way the curve bends, then state it

Over or under

Curve on the interval	Estimate
Bends upward (chords above the curve), e.g. eˣ, sec²x on (−π/2, π/2), 1/x for x > 0	over-estimate
Bends downward (chords below the curve), e.g. ln x, √x	under-estimate

Worked reminder: two strips for ∫₂⁴ ln x dx give ½ × 1 × (ln 2 + 2 ln 3 + ln 4) = 2.14 (3 s.f.). The graph of ln x bends downward, so this is an under-estimate.

Exact values you need at the limits

“Exact value” questions depend on these. Learn them so you never reach for a decimal.

Expression	Value
e⁰, ln 1	1, 0
e^(ln k), e^(2 ln k), e^(−ln k)	k, k², 1/k
ln a − ln b, k ln a	ln(a/b), ln(aᵏ)
sin(π/6), cos(π/3)	½
sin(π/3), cos(π/6)	√3/2
sin(π/4), cos(π/4)	√2/2
tan(π/6), tan(π/4), tan(π/3)	1/√3, 1, √3
sin(π/2), cos(π/2)	1, 0

A lower limit of 0 rarely gives zero: e⁰ = 1 and cos 0 = 1 both leave a term to subtract.

How 2.5 is combined with other topics

Paper 2 questions often link integration to another section of the paper or to Paper 1.

• With 2.2 (logs and exponentials): a curve such as y = e^(2x) − keˣ + c meets the x-axis where a quadratic in eˣ is solved; the limits are then logarithms.
• With 2.3 (trigonometry): writing a sin x + b cos x as R cos(x − α) can turn 1/(a sin x + b cos x)² into a multiple of sec²(x − α).
• With 1.8 (Paper 1 integration): areas under exponential or trig curves, finding a curve from dy/dx and a point, and volumes of revolution, where y² must be formed before integrating.
• With the trapezium rule: an estimate is compared with an exact value found by integration, and you explain the sign of the error from the shape of the curve.

Must-know distinctions

• 1/(ax + b) vs 1/(ax + b)². The first gives (1/a) ln|ax + b|. The second is (ax + b)⁻², a Paper 1 power: −1/(a(ax + b)).
• Integrating vs differentiating sin and cos. d/dx(sin) = cos but ∫ sin = −cos.
• sin² vs cos² identities. sin²x = ½(1 − cos 2x); cos²x = ½(1 + cos 2x).
• Strips vs ordinates. “4 intervals” means 5 y-values.
• Exact vs decimal. “Exact value” means leave ln, e, π and surds. “Estimate” or “3 s.f.” means a decimal.
• Increasing vs bending. Over/under depends on which way the curve bends, not on whether it rises or falls.
• Radians vs degrees. All calculus results here need radians. Check calculator mode before any trig evaluation.

Quick self-test

1. Find ∫ e^(5x − 2) dx.
2. Find ∫ 3/(2 − x) dx.
3. Find ∫ sec²(4x) dx.
4. Find ∫ 6 sin(2x + 1) dx.
5. Find the exact value of ∫₀^(π/2) cos(½x) dx.
6. Find the exact value of ∫₀¹ e^(2x) dx.
7. Find the exact value of ∫₂⁵ 1/(x − 1) dx.
8. Find ∫ cos²x dx.
9. Find ∫ (1 + tan²3x) dx.
10. Find ∫ sin x cos x dx.
11. The trapezium rule is used on the interval 0 ≤ x ≤ 3 with 6 strips. State h and the number of ordinates.
12. Does the trapezium rule over- or under-estimate ∫₁⁴ √x dx? Give a reason.

Answers

1. (1/5) e^(5x − 2) + c
2. −3 ln|2 − x| + c (a = −1)
3. ¼ tan 4x + c
4. −3 cos(2x + 1) + c
5. [2 sin(½x)]₀^(π/2) = 2 sin(π/4) = √2
6. [½ e^(2x)]₀¹ = ½(e² − 1)
7. [ln(x − 1)]₂⁵ = ln 4 − ln 1 = ln 4
8. ½x + ¼ sin 2x + c
9. 1 + tan²3x = sec²3x, so (1/3) tan 3x + c
10. ½ sin 2x integrates to −¼ cos 2x + c
11. h = 0.5 and 7 ordinates
12. Under-estimate: y = √x bends downward, so each chord lies below the curve.

Where marks are usually lost

• Writing ∫ e^(3x) dx = 3e^(3x): multiplying by a instead of dividing.
• Dropping the sign when a is negative, as in ∫ 1/(2 − x) dx or ∫ cos(1 − 4x) dx.
• Giving ∫ sin kx dx as (1/k) cos kx, losing the minus sign.
• Treating 1/(ax + b) as (ax + b)⁻¹ and applying the power rule, which fails at n = −1.
• Using sin²x = ½(1 + cos 2x) (wrong sign) or forgetting to halve the angle factor when integrating cos 2x.
• Evaluating trig limits with the calculator in degrees.
• In the trapezium rule, using n + 1 as the number of strips, or doubling y₀ and yₙ.
• Rounding ordinates to 2 d.p. before adding, so the 3 s.f. answer is wrong.
• Stating “over-estimate” or “under-estimate” with no reason about the shape of the curve.
• Leaving an “exact” answer as a decimal, or as ln 9 − ln 3 instead of ln 3.

Official syllabus

Cambridge International AS & A Level Mathematics 9709 syllabus, for exams in 2026 and 2027 (Version 4), Cambridge University Press & Assessment. Topic 2, Pure Mathematics 2 (for Paper 2): section 2.5 Integration.