## Integral supports the whole curriculum

**Integral A level** covers the whole of the Cambridge International AS/A level Mathematics and Further Mathematics syllabuses. The material is presented in topics, which are further divided into sections.

### Mathematics

- Problem solving
- Problem solving and modelling
- Writing mathematics and proof

- Algebra
- Using and manipulating surds
- Quadratic equations and graphs
- The quadratic formula
- Simultaneous equations
- Inequalities

- Coordinate geometry
- Coordinates and straight lines
- Drawing curves
- The circle

- Sequences and series
- Definitions and notation
- Arithmetic progressions
- Geometric progressions
- Binomial expansions

- Functions and transformations
- The language of functions
- Using transformations

- Differentiation
- Introduction to differentiation
- Using differentiation
- Further applications of differentiation
- The chain rule

- Trigonometry
- Functions and identities
- Equations
- The sine and cosine rules

- Integration
- Introduction to integration
- Finding the area under a curve
- Further integration
- Finding volumes by integration

- Trigonometry
- Trigonometrical functions
- Solving equations
- Circular measure

- Algebra
- Solution of polynomial equations
- The modulus function

- Logarithms and exponentials
- Exponential functions and logarithms
- Modelling curves
- The natural logarithm and exponential functions

- Trigonometry
- Reciprocal trigonometrical functions
- Compound-angle and double-angle formulae
- The forms \(r \cos(\theta \pm \alpha), r \sin(\theta \pm \alpha)\)

- Further algebra
- The general binomial expansion
- Partial fractions

- Differentiation
- The product and quotient rules
- Differentiating natural logarithms and exponentials
- Differentiating trigonometrical functions
- Differentiating functions defined implicitly
- Parametric differentiation

- Numerical solution of equations
- Interval estimation and fixed-point iteration

- Further calculus
- Integration by substitution
- The use of partial fractions in integration
- Integration by parts

- Differential equations
- Forming and solving differential equations

- Vectors
- Vectors in two and three dimensions
- The angle between two vectors
- The vector equation of a line

- Complex numbers
- Working with complex numbers
- Sets of points in an Argand diagram
- The modulus-argument form of complex numbers
- Complex numbers and equations

- Integration
- Integrating other functions
- Numerical integration

- Motion in a straight line
- The language of motion
- Acceleration

- The constant acceleration formulae
- Using the formulae

- Forces and Newton's laws of motion
- Force diagrams and motion

- Applying Newton's second law along a line
- Newton's second law
- Connected objects

- Forces in equilibrium and resultant forces
- Resultant forces
- Newton's second law in two dimensions

- Momentum
- Conservation of momentum

- General motion in a straight line
- Using calculus

- A model for friction
- Modelling with friction

- Energy, work and power
- Energy and work
- Power

- Exploring data
- Data and measures of central tendency
- Measures of spread

- Representing and interpreting data
- Further work with data

- Probability
- Working with probability
- Conditional probability

- Discrete random variables
- Introduction
- Expectation and variance

- The normal distribution
- Using the normal distribution

- Permutations and combinations
- Using permutations and combinations

- Discrete probability distributions
- The binomial distribution
- The geometric distribution

- Continuous random variables
- Probability density functions
- Mean, variance, median and percentiles

- Hypothesis testing using the binomial distribution
- Introducing hypothesis testing
- More about hypothesis testing

- Hypothesis testing and confidence intervals using the normal distribution
- Hypothesis testing using the Normal distribution
- Confidence intervals

- The Poisson distribution
- Introduction to the Poisson distribution
- More about the Poisson distribution

- Linear combinations of random variables
- Combining two or more random variables

- Sampling
- Sampling

### Further Mathematics

- Matrices and transformations
- Introduction to matrices
- Transformations
- Invariance

- Series and induction
- Summing series
- Proof by induction

- Roots of polynomials
- Roots and coefficients

- Rational functions and graphs
- Graphs of rational functions
- Sketching curves related to \(y = f(x)\)

- Polar coordinates
- Polar coordinates and curves
- Finding the area enclosed by a polar curve

- Matrices and their inverses
- The determinant and inverse of a 2x2 matrix
- The inverse of a 3x3 matrix

- Vectors
- The vector equation of a plane
- Points, lines and planes
- The vector product

- Hyperbolic functions
- The hyperbolic and inverse hyperbolic functions

- Matrices
- Simultaneous equations and intersecting planes
- Eigenvalues and eigenvectors
- Powers of square matrices

- Differentiation
- Differentiating further functions
- Maclaurin series

- Integration
- Integrating further functions
- Integration using reduction formulae
- Approximation of areas using rectangles
- Arc lengths and surface areas

- Complex numbers
- de Moivre's theorem
- Applications of de Moivre's theorem

- Differential equations
- Integrating factors
- Second order homogeneous differential equations
- Non-homogeneous differential equations

- Motion of a projectile
- Projectile problems
- The path of a projectile

- Moments of forces
- Rigid bodies
- The moment of a force that acts at an angle

- Centre of mass
- Finding a centre of mass
- Sliding and toppling

- Circular motion
- Motion in a horizontal circle
- Motion in a vertical circle

- Hooke's law
- Elastic strings and springs
- Work and energy

- Linear motion under a variable force
- Differential equations

- Momentum
- Newton's experimental law
- Oblique impacts

- Continuous random variables
- Further probability density functions
- The cumulative distribution function

- Inference using normal and t-distributions
- Using the t-distribution
- Working with two samples

- Chi-squared tests
- The chi-squared test for a contingency table
- Goodness of fit tests

- Non-parametric tests
- Single sample tests
- Paired-sample and two-sample tests

- Probability generating functions
- Using probability generating functions

Each section contains a standard set of resources, including: