IGCSE Additional Mathematics: Integration Practice Questions
Integration reverses differentiation. Increase the index by one and divide by the new index, adding a constant for an indefinite integral. A definite integral evaluates the result at both limits and subtracts.
Topic 16 of Cambridge IGCSE Additional Mathematics 0606 is procedurally simple but unforgiving about the constant of integration and the order of limits. Area questions are the main application. The questions below cover all four cases.
What you need to know for Integration
- Power rule for integrationThe integral of xn is xn+1 divided by (n + 1), plus a constant, valid for every n except minus 1.
- Constant of integrationAn indefinite integral must include plus c. Omitting it loses a mark every time.
- Definite integralEvaluate the integrated expression at the upper limit and subtract its value at the lower limit. No constant is needed.
- Area under a curveThe definite integral between two limits gives the area between the curve and the x-axis, provided the curve does not cross the axis in that interval.
- Integrating a bracketFor (ax + b)n, raise the index by one, divide by the new index, and divide again by a.
IGCSE Additional Mathematics Integration questions and answers
4 exam-style questions written to the 0606 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Find the integral of (6x2 minus 4x + 3) with respect to x.
Show the worked answer
- Integrate term by term, raising each index by one and dividing by the new index.
- 6x2 becomes 6x3 divided by 3 = 2x3.
- minus 4x becomes minus 4x2 divided by 2 = minus 2x2, and 3 becomes 3x.
- Add the constant of integration: 2x3 minus 2x2 + 3x + c.
Evaluate the definite integral of (2x + 1) with respect to x, from x = 1 to x = 3.
Show the worked answer
- Integrate: the integral of (2x + 1) is x2 + x. No constant is needed for a definite integral.
- Evaluate at the upper limit: 32 + 3 = 12.
- Evaluate at the lower limit: 12 + 1 = 2.
- Subtract: 12 minus 2 = 10.
Calculate the area enclosed between the curve y = x2, the x-axis and the lines x = 0 and x = 3.
Show the worked answer
- The area is the definite integral of x2 from 0 to 3.
- Integrate: the integral of x2 is x3 divided by 3.
- Evaluate at the upper limit: 27 divided by 3 = 9. At the lower limit: 0.
- Area = 9 minus 0 = 9 square units. The curve stays above the axis throughout, so no part of the area is negative.
Find the integral of (2x + 1)4 with respect to x.
Show the worked answer
- Raise the index by one: the bracket becomes (2x + 1)5.
- Divide by the new index, 5.
- Divide again by the coefficient of x inside the bracket, which is 2, reversing the chain rule.
- The result is (2x + 1)5 divided by 10, plus c. Check by differentiating: 5 x 2 x (2x + 1)4 divided by 10 = (2x + 1)4.
Common mistakes in this topic
- Omitting the constant of integration.
- Subtracting the limits in the wrong order.
- Forgetting to divide by the inside coefficient when integrating a bracket.
- Treating a negative integral value as an area without checking the curve's position.
- Applying the power rule when the index is minus 1, where it does not hold.
Exam tips
- Check any integration by differentiating the answer. It should return the original expression.
- Write plus c automatically on every indefinite integral before evaluating anything.
- For definite integrals, write the square bracket notation with the limits. It structures the working.
- Sketch the curve before an area question to see whether it crosses the x-axis.
- Upper limit minus lower limit, always in that order.
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Integration FAQs
How do I integrate a power of x?
Raise the index by one and divide by the new index, then add a constant of integration for an indefinite integral. So the integral of x squared is x cubed over 3 plus c. The rule works for every index except minus 1.
Why do I need the constant of integration?
Differentiation destroys constant terms, so infinitely many functions share the same derivative, differing only by a constant. The plus c represents that whole family. It is not needed for a definite integral, because it cancels when the limits are subtracted.
How do I find the area under a curve?
Evaluate the definite integral between the two x values, taking the upper limit value minus the lower limit value. Check first whether the curve crosses the x-axis in that interval, because any section below the axis gives a negative value that must be handled separately.
How do I integrate a bracket raised to a power?
Raise the index by one, divide by the new index, then divide again by the coefficient of x inside the bracket. This reverses the chain rule. Differentiating your answer should return the original expression exactly.
Continue through the IGCSE Additional Mathematics syllabus
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Written to the published Cambridge IGCSE Additional Mathematics (0606) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.