About this chapter
Pure 4 integration is the biggest chapter in this booklet, worth 86 marks: substitution, integration by parts, partial fractions, areas and volumes under parametric curves, and differential equations. The January 2024 paper tested much of it, and its examiners noted the same weakness several times: not linking a later part to the result just found.
The questions cover integration by parts, including twice, substitutions with changed limits, partial fractions, trig integrals, parametric areas, volumes of revolution, separable differential equations, a logistic population model, and exact answers in log form. One volume question needs a numerical method to finish.
The ten questions
- Q1 (5 marks): integrate x sec²x by parts from 0 to π/4, giving an exact aπ + b ln 2.
- Q2 (6 marks): the substitution u = √x turns an integral into a + b ln 2.
- Q3 (8 marks): partial fractions, then an integral from k to 2k equal to a log fixes k.
- Q4 (5 marks): the volume when y = 2 + 1/x is rotated about the x-axis, in exact form.
- Q5 (6 marks): a separable differential equation with cos²y, solved as y = f(x).
- Q6 (11 marks): an area under a parametric curve, its Cartesian equation, and a volume of revolution.
- Q7 (10 marks): integrate e to the power −x times sin x by parts twice, then compare two areas.
- Q8 (12 marks): a logistic model for fish in a lake, solved with partial fractions, with its long-term value and an exact time.
- Q9 (12 marks): a volume of revolution in terms of k, its limit, and an iteration for the k giving 90% of it.
- Q10 (11 marks): integrate x cos 2x by parts, then use it to solve a differential equation.
Key skills tested
Integration by parts. The integral of u dv/dx is uv minus the integral of v du/dx. Let u = x (or x²), but always let u = ln x.
Substitution. Find du/dx, replace dx properly rather than just renaming it du, change the limits, and leave no x in the integral.
Partial fractions. Split first. k/(ax + b) integrates to (k/a) ln|ax + b| + c, so do not lose the 1/a.
Trig integrals. Use cos²x = ½(1 + cos 2x). sec²x integrates to tan x, and tan x to ln|sec x|.
Parametric areas. Integrate y times dx/dt with respect to t, using values of t as the limits.
Volumes of revolution. V is π times the integral of y², so square the whole of y. For a parametric curve, integrate π y² dx/dt with respect to t.
Separable equations. Separate the variables, integrate both sides, and add + c at once. Use the condition to find c, then rearrange.
Modelling. As t grows, e to the power −kt tends to 0, which gives the long-term value. Find times exactly with ln.
Exact answers. Combine logarithms, for example ln 5 − ln(1/3) = ln 15, and give the form asked for, such as a + b ln 2.
Worked example
Question 5 from this chapter. Find the particular solution of dy/dx = (2x + 1) cos²y, with −π/2 < y < π/2, for which y = π/4 when x = 0.
Separate the variables: divide by cos²y to get sec²y dy = (2x + 1) dx.
Integrate both sides: tan y = x² + x + c.
Use x = 0, y = π/4: tan(π/4) = 1, so c = 1. Then tan y = x² + x + 1, so y = arctan(x² + x + 1).
Adding the constant as soon as you integrate, before rearranging, is what the final marks depend on.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2024, Q2(b): integrating three partial fractions and combining the logs into one, as in Q3.
- January 2024, Q5: integration by parts twice, then a separable differential equation that reuses that result, as in Q7 and Q10.
- January 2024, Q7: a given substitution, then a volume of revolution using it, as in Q2, Q4 and Q6.
Where marks are lost
- No + c. In January 2024 many solved the differential equation correctly but left out the constant, and lost the final mark.
- Signs in the second round of parts. Errors were usually in the second use of integration by parts, particularly when the two stages were worked separately and then combined.
- Jumping to the answer. In the January 2024 substitution many lost two marks by going straight to the printed result with no integral in u and no new limits.
- Not linking to the earlier part. Many started integration by parts again instead of using the result they had just found.
Common questions
Which part is u in integration by parts?
Usually the power of x, because it gets simpler when differentiated. The exception is ln x, which should always be u.
Do I have to change the limits in a substitution?
Yes, if you finish in terms of u. The January 2024 examiners expected the limits in terms of u to be shown.
When do I add + c in a differential equation?
As soon as you integrate both sides, before rearranging or using the given condition.
Where this chapter leads
- Pure 4 Chapter 7: Vectors: the final chapter of Pure 4.
- Pure 3 Chapter 8: Numerical Methods: the iteration used to finish volume questions such as Q9.