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

Introduction to integration

What is integration?

• Integration is the inverse (or '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 its 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

    • In this case it is integrate with respect to

  •  is the function to be integrated (sometimes called the integrand)

• The antiderivative is sometimes denoted by 

  • Then there’s no need to keep writing the whole integral; refer to it as 

•  may also be called the indefinite integral of

• notation can also be used

  • So instead of integrating to find its antiderivative

  • you can think of integrating to find an expression for its antiderivative

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

  • 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

  • Each one corresponds to a different possible value of