Pure Maths
A-Level Integration
Integration reverses differentiation and measures accumulated change, most visibly as the area between a curve and an axis.
What you need to know
- Integrating powers of x and adding the constant of integration
- Definite integrals and area under a curve
- Area between two curves
- Integration by substitution and by parts
Key concepts
- Indefinite integrals always need + c.
- Area below the x-axis gives a negative definite integral, so split the interval.
- For area between curves, integrate (upper − lower).
Common mistakes
- Omitting the constant of integration
- Substituting limits in the wrong order
- Treating a negative integral as a negative area
Example question
Find .
Show explanation
Increase the power by one and divide: 6x³/3 = 2x³, plus the constant of integration.
Practise Integration
4 original questions with worked explanations, mixed across easy, medium and challenging difficulty.
Integration questions students ask
- Why does my area come out negative?
- The curve is below the axis on part of the interval. Integrate each region separately and take the modulus before adding.