Integration

Reverse differentiation, find accumulated change and calculate an area.

Choose the skill you need. Move the boundaries and compare curves yourself, keep the written method in view, and practise with hints and full solutions.

Integration: powers and polynomialsIntegrate powers, polynomials, roots and reciprocals. Rewrite expressions, include the constant of integration and differentiate your answer to check it.
Finding the constant of integrationRecover a curve from its derivative and a known point. Find the integration constant, use initial conditions and interpret motion models and repeated integration.
Definite integralsEvaluate definite integrals with upper minus lower, exact fractions and surds. Solve unknown-limit and parameter questions, and check domains before applying the rule.
Area under a curve using integrationFind areas above and below the x-axis, split regions at crossings and distinguish total area from a signed integral. Includes repeated roots, transformations and equal-area problems.
Area between curves using integrationFind area between a curve and a line or two curves. Solve intersections, use upper minus lower, split changing boundaries and combine integrals with triangles and trapeziums.

Useful foundations

Review the differentiation power rule, index laws or simultaneous equations when those steps need a refresher.