Introduction to Integration (Cambridge International IGCSE Further Maths)

Introduction to integration

What is integration?

• Integration is the opposite to differentiation
• Integration is the process of finding the expression of a function from an expression of the derivative(gradient function)

What is the fundamental theorem of calculus?

• The Fundamental Theorem of Calculus states that integration is the inverse process of differentiation

  • This form of the Theorem relates to Indefinite Integration
  • An alternative version of the Fundamental Theorem of Calculus involves Definite Integration

What is the constant of integration (+c)?

• When differentiating y, constant terms ‘disappear’

  • for constants y= c, $\frac{dy}{dx}=0$
  • graphs of constants are horizontal lines and so have gradient of 0

• Integrating $\frac{dy}{dx}$, to get y, cannot determine the constant

  • To acknowledge this constant, “+ c” is used
  • c is called the constant of integration

What is the notation for integration?

• An integral is normally written in the form

$\text{integral} f ( x ) \mathrm{d} x$

• the large operator integral means “integrate”

  • “$dx$” indicates which variable to integrate with respect to
  • $f(x)$ is the function to be integrated

• If it has more than one term the function to be integrated (called the integrand) should be in brackets

  • “Integrate” -– “all of (…)” -– “with respect to x”

Worked Example