Paper 3 integration asks students to choose a method: substitution, by parts, partial fractions, a trigonometric identity, or a standard form. The reports show that the choice is often right and the execution is where marks go. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Signs and coefficients. The most repeated problem. Typical examples are a missing minus sign when integrating a sine, e^(2u) integrated as 2e^(2u), losing a 5 or a 2 in a substitution, and wrong coefficients in terms involving trigonometric functions.
- Integration by parts used once. In a hard question, many students applied the formula once and did not manage to complete the integration. Others used parts where a simpler identity or substitution would have worked.
- Substitution errors. Using an alternative substitution to the one the question specified, losing a factor in du/dx, an error in the double angle formula, and arithmetic or algebraic slips when substituting. When a substitution is given, the integrand must be written completely in terms of the new variable. Reports note many errors in substituting after du/dx was found correctly.
- Trigonometric integrals. A few students recognised that sin²2x should be rewritten as (1 − cos 4x)/2, but it was more common to see sin³2x or cos³2x. tan x was replaced with sec²x − 1, often with a sign error, and the student integrated the tan x part but not the tan x sec²x part. Integrating sec θ tan θ produced a complicated expression with logarithms.
- Standard forms. In an integral with a numerator containing x, students ignored the numerator and assumed the answer was of the form p tan⁻¹ q, or knew it was a logarithm but kept x as a coefficient.
- Algebraic fractions. Some students saw that the integrand could be written as a quotient plus a remainder over the denominator, but then dropped the denominator. In another question, students tried a variety of approaches to a fraction such as (x + 2)/(x + 1), most of them incorrect, instead of dividing or rewriting the numerator.
- Logarithm handling. Taking ln of both sides when it was unnecessary, errors with brackets that made it hard to see whether a logarithm law had been applied correctly, and incorrect combination of logarithms at the end.
- Not finishing. Students did not substitute the limits and equate the answer to 4, or gave only the positive value as the final answer.
- Working from a wrong integral. Students who had an incorrect integral often tried to reverse-engineer the given form of the answer. One report says their time is better spent reviewing the solution to find the error.
- Correct integral from wrong working. An apparently correct integral obtained from wrong working, such as starting with the wrong numerator, earns no credit.
- Brackets and presentation. Omitting brackets was one of the commonly recorded errors, and the reports remind students to include essential brackets and present work clearly.
- Calculator calls. Calculator-evaluated integrals earn no credit for method.
What examiners advise
- Decide the method before you start. The key messages for this paper say to gauge the method that best suits a problem, as there are often several approaches.
- Follow the substitution you are given. If a question says to use a substitution, use that one.
- Quote and apply the identity you use for trigonometric integrals.
- Write out the remainder and quotient when the integrand is an improper fraction.
- Show the limits as you substitute them.
- Differentiate your answer if you have time.
Habits that help
- Write down u, du/dx and the new limits at the very start of a substitution question.
- Keep a list of the standard integrals and their derivatives and check each against it.
- For by-parts questions, write u, v′, u′ and v before assembling.
Questions students ask
How do I choose the integration method in Paper 3?
Decide before you start. The key messages say to gauge the method that best suits the problem, as there are often several approaches. Students used parts where a simpler identity or substitution would have worked.
What if a question gives a substitution?
Use that one. Reports record students using an alternative substitution, losing a factor in du/dx, or making slips with the double-angle formula.
How do I integrate sin²2x?
Rewrite it as (1 − cos 4x)/2 first. Only a few students did. Quote the identity you use, and use the formula book when needed.
How do I integrate (Ax + B)/(x² + 2)?
Split the numerator. Students ignored it, assumed p tan⁻¹ q, or knew it was a logarithm but kept x as a coefficient.
What if my integral does not match the given answer?
Review your solution to find the error. Reports say that students who reverse-engineer the printed answer waste time, and a correct-looking integral from wrong working earns no credit.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.