A Level Maths 9709 · Paper 3 Pure Mathematics 3 · 3.5 (also Paper 2) · ~8 min read
Further integration
Integrating eˣ, 1/x, trig functions and 1/(a² + x²), using identities and partial fractions, the f′(x)/f(x) pattern, integration by parts, a given substitution, and the trapezium rule with over- and under-estimates.
Standard integralsTrig identitiesPartial fractionsf′(x)/f(x)Integration by partsSubstitutionTrapezium ruleOver/under-estimates
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Paper tag = where it is examinedFORMULA LIST (MF19) given in the examM1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end
In 30 seconds
- Reverse the derivative list: ∫eax+b dx = (1/a)eax+b; ∫1/(ax + b) dx = (1/a) ln ǀax + bǀ; ∫1/(a² + x²) dx = (1/a) tan⁻¹(x/a).
- Match the method to the shape: a product of x and a function → parts; f′(x)/f(x) → ln ǀf(x)ǀ; a fraction with a factorised denominator → partial fractions; sin² or cos² → double-angle identity.
- Substitution: change every part, x, dx and the limits, into u.
- Trapezium rule: h/2 × [first + last + 2(middle)], with h = (b − a)/n. It over-estimates for a curve bending upwards and under-estimates for one bending downwards.
By the end you can
- Integrate eˣ, 1/x, sin, cos, sec² and 1/(a² + x²) with linear arguments.
- Use trig identities and partial fractions before integrating.
- Recognise and integrate f′(x)/f(x).
- Integrate by parts and by a given substitution, including definite integrals.
- Use the trapezium rule and say whether it gives an over- or under-estimate.
1Standard integrals
The list
| f(x) | ∫f(x) dx |
|---|---|
| eax + b | (1/a) eax + b + c |
| 1/(ax + b) | (1/a) ln ǀax + bǀ + c |
| sin(ax + b) | −(1/a) cos(ax + b) + c |
| cos(ax + b) | (1/a) sin(ax + b) + c |
| sec²(ax + b) | (1/a) tan(ax + b) + c |
| 1/(a² + x²) | (1/a) tan⁻¹(x/a) + c |
- These are in the formula list FORMULA LIST (MF19). The usual error is the factor 1/a: always check by differentiating your answer.
- The modulus in ln ǀxǀ matters when x can be negative. For definite integrals, the limits must not cross the asymptote.
Worked example
Find the exact value of ∫₁³ 1/(2x − 1) dx.
- [½ ln(2x − 1)]₁³ M1 A1
- = ½ ln 5 − ½ ln 1 = ½ ln 5 A1
Worked example
Find the exact value of ∫₀² 1/(4 + x²) dx.
- a = 2: [½ tan⁻¹(x/2)]₀² M1 A1
- = ½ tan⁻¹1 = ½ × π/4 = π/8 A1
Common mistakeWriting ∫e3x dx = 3e3x. Integrating divides by 3, differentiating multiplies.
2Using identities
sin², cos², tan²
cos²x ≡ ½(1 + cos 2x) ; sin²x ≡ ½(1 − cos 2x) ; tan²x ≡ sec²x − 1 LEARN IT
from cos 2x and 1 + tan²x ≡ sec²xWorked example
Find the exact value of ∫₀π/4 cos²x dx.
- ½∫₀π/4 (1 + cos 2x) dx = ½[x + ½ sin 2x]₀π/4 M1 A1
- = ½(π/4 + ½) = π/8 + ¼ A1
- Products like sin x cos x: use ½ sin 2x, giving −¼ cos 2x.
Common mistakeIntegrating sin²x as sin³x/3. That is the chain rule backwards and only works if a cos x factor is present.
3Partial fractions and f′/f
Splitting first
- Factorised denominator? Split into partial fractions (Algebra), then each piece gives a log.
Worked example
Show that ∫₂⁴ (5x + 1)/[(x − 1)(x + 2)] dx = ln(243/8).
- (5x + 1)/[(x − 1)(x + 2)] = 2/(x − 1) + 3/(x + 2) B1
- [2 ln(x − 1) + 3 ln(x + 2)]₂⁴ = (2 ln 3 + 3 ln 6) − (0 + 3 ln 4) M1 A1
- = ln 9 + ln(6/4)³ = ln(9 × 27/8) = ln(243/8) A1 AG
The f′(x)/f(x) pattern
∫ f′(x)/f(x) dx = ln ǀf(x)ǀ + c
the top is (a multiple of) the derivative of the bottom- Examples: ∫2x/(x² + 1) dx = ln(x² + 1); ∫tan x dx = ∫sin x/cos x dx = −ln ǀcos xǀ = ln ǀsec xǀ.
Worked example
Find the exact value of ∫₀π/3 tan x dx.
- [−ln cos x]₀π/3 M1
- = −ln ½ + ln 1 = ln 2 A1
Common mistakeUsing ln for 1/(x² + 1). The top must be the derivative of the bottom; 1/(x² + 1) gives tan⁻¹x instead.
4Integration by parts
∫u dv/dx dx = uv − ∫v du/dx dx
∫u (dv/dx) dx = uv − ∫v (du/dx) dx FORMULA LIST (MF19)
from the product rule, integrated- Choosing u: pick the part that gets simpler when differentiated. Priority: ln x first, then powers of x, then eˣ or trig.
- ln x on its own: write it as 1 × ln x with u = ln x, dv/dx = 1.
- x² eˣ needs parts twice.
Worked example
Find the exact value of ∫₀π/2 x sin 2x dx.
- u = x, dv/dx = sin 2x ⇒ v = −½ cos 2x M1
- = [−½x cos 2x]₀π/2 + ½∫₀π/2 cos 2x dx A1
- = −½(π/2)(−1) + ½[½ sin 2x]₀π/2 = π/4 + 0 M1
- = π/4 A1
Worked example
Find ∫₁e x² ln x dx.
- u = ln x, dv/dx = x² ⇒ v = x³/3 M1
- = [x³ ln x/3]₁e − ∫₁e x²/3 dx A1
- = e³/3 − [x³/9]₁e = e³/3 − e³/9 + 1/9 M1
- = (2e³ + 1)/9 A1
Worked example
Find ∫₁e ln x dx.
- u = ln x, v = x: [x ln x]₁e − ∫₁e 1 dx M1
- = e − (e − 1) = 1 A1
Common mistakeSign errors with the minus in uv − ∫v du. Write the line out in full and bracket v du/dx.
5Integration by substitution
Change everything to u
- Steps: (1) write du/dx and make dx the subject; (2) replace x everywhere; (3) change the limits; (4) integrate in u. Never mix x and u in one integral.
Worked example
Use the substitution u = √x to find the exact value of ∫₁⁴ 1/(x + √x) dx.
- x = u², dx = 2u du; limits x = 1 → u = 1, x = 4 → u = 2 M1
- ∫₁² 2u/(u² + u) du = ∫₁² 2/(u + 1) du A1
- = [2 ln(u + 1)]₁² = 2 ln 3 − 2 ln 2 M1
- = 2 ln(3/2) A1
Worked example
Use u = sin x to find ∫₀π/2 sin³x cos x dx.
- du = cos x dx; limits 0 and 1 M1
- ∫₀¹ u³ du = [u⁴/4]₀¹ = ¼ A1
Common mistakeNot changing the limits, or leaving dx in place of 2u du. The 2024 Paper 3 report flagged both.
6The trapezium rule
Estimating an area
∫ₐᵇ y dx ≈ (h/2)[y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)], h = (b − a)/n FORMULA LIST (MF19)
n strips means n + 1 ordinates- Over or under? Convex curve (bending upwards, like eˣ) → over-estimate. Concave curve (bending downwards, like √x) → under-estimate. Justify with a sketch or the sign of d²y/dx².
Worked example
Use the trapezium rule with two intervals to estimate ∫₀¹ ex² dx, and say whether it is an over- or under-estimate.
- h = 0.5; y = 1, e0.25 = 1.2840, e = 2.7183 B1
- 0.25 × [1 + 2.7183 + 2(1.2840)] = 1.57 M1 A1
- the curve bends upwards on [0, 1], so it is an over-estimate B1
Common mistakeCounting strips as ordinates: 4 intervals need 5 y-values. And radians for any trig function.
Linking questionThe derivatives these reverse are in Further differentiation; Paper 1 integration and areas are in Integration; and these techniques solve Differential equations.
Test yourself — cover the answers first
Quick check
- Integrate e−2x.
- Integrate 1/(9 + x²).
- Which method for ∫x eˣ dx?
- Integrate 3x²/(x³ + 4).
- Write sin²x in terms of cos 2x.
Exam-style
- Find the exact value of ∫₀¹ x e−x dx.
- Find the exact value of ∫₀π/2 x cos x dx.
- Show that ∫₀ln 2 e3x dx = 7/3.
Show answers
- −½e−2x + c
- ⅓ tan⁻¹(x/3) + c
- by parts, u = x
- ln ǀx³ + 4ǀ + c
- ½(1 − cos 2x)
- by parts: [−x e−x]₀¹ + ∫₀¹ e−x dx = −e−1 + 1 − e−1 = 1 − 2/e
- [x sin x]₀π/2 − ∫sin x dx = π/2 + [cos x]₀π/2 = π/2 − 1
- [⅓e3x]₀ln 2 = ⅓(8 − 1) = 7/3
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