Integration

Scheme of work: Year 12 A-Level: Pure 1: Integration

Prerequisite Knowledge

  • From GCSE
    • Calculate with roots, and with integer and fractional indices
    • Simplify and manipulate algebraic expressions
    • Simplifying expressions involving sums, products and powers, including the laws of indices
    • Calculate exactly with surds
    • Simplify surd expressions involving squares and rationalise denominators
  • From Pure Year 1
    • Find the gradient of curves from first principals and understand this as differentiation.
    • Differentiate, for rational values of n, and related constant multiples, sums and differences

Success Criteria

  • Understand integration as the reverse of differentiation.
  • Use the gradient function and a point on the curve to find the equation of a curve.
  • Understand that indefinite integration involves an arbitrary constant.
  • Evaluate definite integrals.
  • Use definite integration to find the area enclosed within a curve and x-axis.
  • Understand that area below the x-axis is negative.

Key Concepts

  • Integration as the reverse of differentiation

If dy/dx = xn, then

y=\frac{1}{n+1} x^{n+1}+c, n \neq-1
  • Indefinite Integration
\int k f(x) d x=h \int f(x) dx
\int(f(x)+g(x)) d x=\int f(x) d x+\int g(x) d x
  • Definite Integration of the form xn
\int_{a}^{b} x^{n} d x=\left[\frac{x^{n+1}}{n+1}\right]_{a}^{n}=\frac{b^{n+1}-a^{n+1}}{n+1}, n \neq 1
  • Area Under a Curve

The value of the definite integral represents the area enclosed within the x-axis, the function and the two limits.

When a graph goes below the x-axis, the corresponding values of y are negative, so the area becomes negative.

To find the area where some parts of the curve are above the axis and others are below it, you need to separate the integrals so that the negative and positive values do not cancel each other out.

Common Misconceptions

  • Some students attempt to answer exam questions using the integral function on their calculator, which results in zero marks awarded.
  • Poor notation, namely dropping the dx part of the integral, can result in lost marks.
  • When asked to integrate questions of the form below, some students attempt to integrate the numerator and denominator separately.
\int\left(\frac{2 x^{3}-5}{x^{2}}\right) d x
  • Some students forget to include the ‘c’ term when evaluating an indefinite integral.
  • Integrals must be fully simplified, as marks are lost for not leaving fractions in their simplest form. Some students also include the integral sign as part of their final answer.
  • While most students know to separate integrals when the curve goes from above the axis to below it, some add to the two areas, not knowing the area below the axis is negative.

Integration Resources

Mr Mathematics Blog

Planes of Symmetry in 3D Shapes

Planes of Symmetry in 3D Shapes for Key Stage 3/GCSE students.

Use isometric paper for hands-on learning and enhanced understanding.

GCSE Trigonometry Skills & SOH CAH TOA Techniques

Master GCSE Math: Get key SOH-CAH-TOA tips, solve triangles accurately, and tackle area tasks. Ideal for students targeting grades 4-5.

Regions in the Complex Plane

Explore Regions in the Complex Plane with A-Level Further Maths: inequalities, Argand diagrams, and geometric interpretations.