Introduction to Integration (DP IB Analysis & Approaches (AA)): Revision Note

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

Integrating Powers of x

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