At a glance
- Cambridge 9709
- P1 1.8; P2 2.5; P3 3.5 and 3.8
- Edexcel IAL
- P1 to P4 (WMA11 to WMA14)
- Hardest techniques
- Parts, substitution, partial fractions
- Never forget
- + c on indefinite integrals
Where integration sits in each specification
| Board and paper | Content |
|---|---|
| Cambridge 9709 Paper 1 (1.8) | Integrate (ax + b)^n for n not equal to -1, find the constant of integration, definite integrals including simple improper ones, area between curves and lines, volume of revolution about an axis |
| Cambridge 9709 Paper 2 (2.5) | e^(ax + b), 1/(ax + b), sin, cos and sec^2 of (ax + b), double-angle identities in integration, the trapezium rule |
| Cambridge 9709 Paper 3 (3.5, 3.8) | Paper 2 content plus 1/(x^2 + a^2), kf'(x)/f(x), partial fractions, integration by parts, a given substitution, first-order differential equations |
| Edexcel IAL P1 and P2 | Indefinite integration of x^n (P1); definite integrals, area under a curve, trapezium rule (P2) |
| Edexcel IAL P3 and P4 | e^x, 1/x and trig functions, integration by recognition (P3); volumes of revolution, substitution, parts, partial fractions, differential equations, parametric areas (P4) |
Cambridge students sit Paper 2 for AS only or Paper 3 for the full A Level. Formula lists are provided, but the standard integrals still need to be recognised quickly.
The key ideas
For powers, add one to the power and divide by the new power: the integral of x^n is x^(n+1)/(n+1) + c, for any n except -1. For a linear inside, divide by the coefficient of x as well: the integral of (2x + 3)^4 is (2x + 3)^5/10 + c. The exception n = -1 gives a logarithm: the integral of 1/(ax + b) is (1/a) ln|ax + b| + c.
A definite integral gives the signed area between the curve and the x-axis, so a region below the axis comes out negative. To find the area between two curves, integrate (top curve minus bottom curve) between the x-coordinates where they meet. A volume of revolution about the x-axis is π times the integral of y^2.
For harder integrands, pick the technique from the shape. A product of two different types, such as x e^(2x) or x sin x, suggests integration by parts: the integral of u dv/dx is uv minus the integral of v du/dx, with u chosen as the part that gets simpler when differentiated (ln x is the exception, always take u = ln x). A rational function with a factorised denominator suggests partial fractions. A numerator that is a multiple of the derivative of the denominator, such as 2x/(x^2 + 1), integrates to a logarithm. Powers of sin and cos usually need an identity such as sin^2 x = (1 - cos 2x)/2.
Worked example 1: the area between y = 9 - x^2 and y = 5
- Find where they meet: 9 - x^2 = 5 gives x^2 = 4, so x = -2 and x = 2.
- Between these values the parabola is on top, so integrate (9 - x^2) - 5 = 4 - x^2.
- Integrate: the integral of (4 - x^2) is 4x - x^3/3.
- Substitute the limits: (8 - 8/3) - (-8 + 8/3) = 16/3 + 16/3 = 32/3.
- Answer: the area is 32/3, about 10.7 square units. A sketch confirms a region of width 4 and height 4 that is less than a full 4 by 4 square.
Worked example 2 (Cambridge P3, Edexcel P4): by parts, find the integral of x e^(2x) from 0 to 1
- Choose u = x (it becomes 1 when differentiated) and dv/dx = e^(2x), so du/dx = 1 and v = (1/2)e^(2x).
- Apply the formula: integral = (x/2)e^(2x) minus the integral of (1/2)e^(2x).
- Integrate the remaining part: (1/2)e^(2x) integrates to (1/4)e^(2x).
- So the indefinite integral is (x/2)e^(2x) - (1/4)e^(2x) + c.
- Apply the limits: at x = 1 it is e^2/2 - e^2/4 = e^2/4; at x = 0 it is 0 - 1/4. Subtract: e^2/4 + 1/4 = (e^2 + 1)/4, about 2.097.
Worked example 3 (Cambridge P3, Edexcel P4): partial fractions, find the integral of (x + 5)/((x - 1)(x + 2)) from 2 to 3
- Write (x + 5)/((x - 1)(x + 2)) = A/(x - 1) + B/(x + 2), so x + 5 = A(x + 2) + B(x - 1).
- Put x = 1: 6 = 3A, so A = 2. Put x = -2: 3 = -3B, so B = -1.
- Integrate: 2 ln(x - 1) - ln(x + 2).
- Apply the limits: (2 ln 2 - ln 5) - (2 ln 1 - ln 4) = 2 ln 2 - ln 5 + ln 4.
- Combine with log laws: ln 4 + ln 4 - ln 5 = ln 3.2, about 1.163.
Common mistakes that cost marks
- Dropping + c on an indefinite integral, which loses the final accuracy mark.
- Multiplying by the inside coefficient instead of dividing: the integral of cos 3x is (1/3) sin 3x, not 3 sin 3x.
- Treating the integral of 1/x as x^0/0. It is ln|x|.
- Adding areas above and below the axis without splitting the integral, so they cancel.
- In integration by parts, choosing u = e^x and making the integral harder.
- In substitution, forgetting to change dx to du and the limits to u values.
- Integrating a product as the product of the integrals.
Exam technique and how a tutor helps
Integration questions are usually worth 5 to 10 marks and often follow on from differentiation or partial fractions earlier in the question. Write the integrated expression in square brackets with the limits before substituting, so the examiner can award the method mark for the integration even if the arithmetic slips. Give exact answers (ln 3.2, (e^2 + 1)/4) when the question says exact, and decimals only when asked.
When a question gives a substitution, use it exactly as given, change the limits, and show the new integral in u before integrating. For volumes, remember the π and square y before integrating.
In one-to-one lessons a tutor gives the student a mixed set of integrands and asks only 'which technique and why' before any integration happens. That habit of choosing first fixes most of the lost time in the exam. The tutor then watches full solutions on the shared whiteboard to catch the sign and coefficient slips that cost the accuracy marks.
Self-check: can you do these?
- Find the integral of (2x + 3)^4. (Answer: (2x + 3)^5/10 + c)
- Find the integral of cos 3x. (Answer: (1/3) sin 3x + c)
- Find the integral of sin^2 x from 0 to π/2. (Answer: π/4)
- Find the integral of ln x by parts. (Answer: x ln x - x + c)
- Find the integral of 2x/(x^2 + 1). (Answer: ln(x^2 + 1) + c)
Common questions
Which A-Level Maths paper tests integration?
In Cambridge 9709 basic integration is in Paper 1, standard functions and the trapezium rule are in Paper 2, and parts, substitution, partial fractions and differential equations are in Paper 3. In Edexcel IAL integration runs through P1 to P4, with parts, substitution and volumes in P4.
Will the exam tell me which substitution to use?
In Cambridge 9709 Paper 3 the syllabus says 'use a given substitution', so the substitution is provided. Edexcel IAL P4 covers simple cases of integration by substitution, so practise spotting a sensible substitution as well as using a given one. Either way, change dx and the limits correctly.
How do I know when to use integration by parts?
Use it when the integrand is a product of two different types of function, such as a polynomial times an exponential or a trig function, or for ln x on its own. Choose u as the part that simplifies when differentiated, except that ln x is always u.
Why is my area answer negative?
Because part of the region is below the x-axis, where the definite integral is negative. Split the integral at the root, take the size of each part, and add them.
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