Indefinite Integrals (College Board AP® Calculus AB)

Indefinite integrals

What is an indefinite integral?

• The indefinite integral of a function is denoted by

  • is the mathematical symbol for 'integrate'

    • When we find the indefinite integral of we are integrating the function

  • The in says that we are integrating 'with respect to x'

• The indefinite integral is defined by

  • 

    • where is a function such that

      • is known as an antiderivative of

    • and is any constant

      • is known as the constant of integration

• Integration is the inverse of differentiation

  • Integrating gives you ()

  • And differentiating () gives you

• Note that the indefinite integral of a function of

  • is another function of

Why do I need the constant of integration +C?

• To be an antiderivative of , the function must satisfy

• Say you found an for which that is true

  • Add a constant to that

    • 

  • And then differentiate (remember that the derivative of a constant is zero)

    • 

  • I.e. if is an antiderivative of

    • then is also an antiderivative of

• This shows that there is no unique antiderivative of a function

  • There is only a family of antiderivatives

    • each differing from the others by a constant value

  • The graphs of these antiderivatives are all vertical translations of each other