Integration questions in Additional Math include indefinite and definite integrals, finding a curve from its gradient, and areas. The reports show that the usual errors are in the process of integrating, in the limits, and in the plan for areas. 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
- Not recognising a logarithm. Some candidates did not realise that integrating 1/x (or a power of −1) produces a logarithmic function, could not make a start, or wrongly wrote ln(...) for a power that does not give a logarithm (such as (10x − 1)⁻⁵).
- The coefficient. Multiplying by a number when they should divide, omitting to divide (10x − 1)⁻⁵ by −5 and by 10, giving ln(4x − 3) with an incorrect multiple, or writing −3 ln(4 − 3x) when the coefficient should be different. Reports also mention a missing bracket around the argument of the logarithm.
- The constant of integration. Stating the correct integral and omitting the constant, or making sign errors when finding the second constant.
- Trigonometric functions. Treating sec² (x/3) as a product or as a power of x, rewriting it in a way that made no progress, altering the argument of a trigonometric function, differentiating instead of integrating, or multiplying the trigonometric function by a function of x instead of a constant. Others made slips with integrals such as sin(2x − 3).
- Expanding first. Errors in expanding the brackets, such as omitting the cross terms, or not simplifying sufficiently before integrating.
- Limits. Not showing the substitution of the limits (offering only decimal values), substituting limits before integrating, using limits that are values of y when integrating with respect to x, and sign errors or incorrect limits when finding the area of a triangle by integration.
- Areas. Finding the area under a curve and not subtracting it from π (or from the area it is part of), not identifying which areas need to be added, not relating the area to the previous part, and leaving out the area of a triangle or rectangle.
- Stationary point and gradient. Substituting the coordinates of a stationary point into the original function (y = 36/4) instead of recognising that the stationary point condition applies to the derivative. In another question, students differentiated the equation of the curve to find the x-coordinate, instead of using the gradient they were given.
- Notation. Omitting dx where needed meant that many candidates did not offer any solution of value.
- Accuracy. Premature approximation, often when finding the area of the rectangle, and leaving ln 256 as b along with arithmetic errors when adding fractions.
- Final manipulation. Many could not simplify a logarithmic equation sufficiently to solve for a.
- Differentiating by mistake. The reports say common errors in integration suggested that differentiation may have taken place.
What examiners advise
- Show the substitution of the limits and every other step. The rubric says that no marks will be given for unsupported answers from a calculator.
- Write dx and the correct notation, and "+ c" for an indefinite integral.
- For a log integral, check by differentiating your answer.
- Make a plan for areas: say which areas to add or subtract, and what the limits are.
- Use the information in the question (such as a given gradient) in the most direct way.
- Check that the substitution of the stationary point coordinates uses the derivative for the gradient condition.
Habits that help
- Differentiate your answer to every integral. It should return the expression you started with.
- Write the limits next to the integral sign, and say whether they are x or y values.
- For an area between a curve and a line, write "top − bottom" before you integrate.
Questions students ask
What is the integral of 1/x?
A logarithm: ln x, plus a constant. Some candidates did not realise integrating 1/x, or a power of −1, produces a logarithmic function and could not make a start.
How do I integrate something like (10x − 1)⁻⁵?
Divide by the new power and by the coefficient of x: by −4 and by 10 here. Students multiplied when they should have divided or omitted to divide by both.
Do I always need + c?
For an indefinite integral, yes. Reports record the correct integral with the constant omitted, and sign errors when finding a second constant.
How should I show the limits?
Show the substitution of the limits and every other step. Offering only decimal values loses the method mark. Do not substitute limits before integrating, and use limits that match the variable.
How do I plan an area question?
Say which areas to add or subtract and what the limits are. Students found the area under a curve and did not subtract it from the area it is part of.
Why is dx important?
Omitting dx where needed meant that many candidates did not offer any solution of value. Also write + c for an indefinite integral.
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.