5.3.2 Introduction to Integration

What is integration?

• Integration is the opposite to differentiation
  • Integration is referred to as antidifferentiation
  • The result of integration is referred to as the antiderivative
• Integration is the process of finding the expression of a function (antiderivative) from an expression of the derivative (gradient function)

What is the notation for integration?

• An integral is normally written in the form

• • the large operator  means “integrate”
  • “” indicates which variable to integrate with respect to
  •  is the function to be integrated (sometimes called the integrand)
• The antiderivative is sometimes denoted by 
  • there’s then no need to keep writing the whole integral; refer to it as 
•  may also be called the indefinite integral of

What is the constant of integration? 

• Recall one of the special cases from Differentiating Powers of x
  • If then
• This means that integrating 0 will produce a constant term in the antiderivative
  • a zero term wouldn’t be written as part of a function
  • every function, when integrated, potentially has a constant term
• This is called the constant of integration and is usually denoted by the letter 
  • it is often referred to as “plus ”
• Without more information it is impossible to deduce the value of this constant
  • there are endless antiderivatives, , for a function

How do I integrate powers of x?

• Powers of are integrated according to the following formulae:
  • If then where and is the constant of integration

• • This is given in the formula booklet

• If the power of is multiplied by a constant then the integral is also multiplied by that constant
  • If then where and is a constant and is the constant of integration
•  notation can still be used with integration
• Note that the formulae above do not apply when as this would lead to division by zero
• Remember the special case:
  • 
    • e.g.  
  • This allows constant terms to be integrated
• Functions involving roots will need to be rewritten as fractional powers of first
  • eg. If then rewrite as and integrate
• Functions involving fractions with denominators in terms of will need to be rewritten as negative powers of first
  • e.g.  If then rewrite as and integrate    

• The formulae for integrating powers of apply to all rational numbers so it is possible to integrate any expression that is a sum or difference of powers of
  • e.g.  If then
             
• Products and quotients cannot be integrated this way so would need expanding/simplifying first
  • e.g.  If then

What might I be asked to do once I’ve found the anti-derivative (integrated)?

• With more information the constant of integration,, can be found
• The area under a curve can be found using integration