5.2.2 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 formula:
  • 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 ,   is a constant and  is the constant of integration
• This is also given in the formula booklet
•  notation can still be used with integration
• Note that the formulae above do not apply when  as this would lead to division by zero
• Don’t forget the special case:
  • 
    • e.g.  
  • This allows constant terms to be integrated
• 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

How do I integrate sums and differences of powers of x?

• The formulae for integrating powers of  apply to all integers 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 in this way so would need expanding/simplifying first
  • e.g. If  then