Integration

Integration is the reverse of differentiation. It finds a function from its gradient, and definite integrals find the area under a curve.

Key facts

  • ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c (n ≠ −1).
  • Don’t forget + c for indefinite integrals.
  • Definite integral: ∫_a^b f(x) dx = F(b) − F(a).
  • Areas below the x-axis come out negative.
  • ∫ 1/x dx = ln|x| + c, ∫ eˣ dx = eˣ + c.

Worked examples

Example 1

Find ∫ (6x² − 4x + 3) dx.

  1. Add one to each power and divide by the new power.
  2. 2x³ − 2x² + 3x + c

Answer: 2x³ − 2x² + 3x + c

Example 2

Find the area under y = x² + 1 between x = 0 and x = 3.

  1. ∫_0^3 (x² + 1) dx = [x³/3 + x]_0^3
  2. = (9 + 3) − 0 = 12

Answer: 12