Indefinite Integrals (College Board AP® Calculus AB): Study Guide

Indefinite integrals

What is an indefinite integral?

• The indefinite integral of a function $f$ is denoted by $\int f\left(x\right)dx$

  • $\int$ is the mathematical symbol for 'integrate'

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

  • The $x$ in $dx$ says that we are integrating $f\left(x\right)$ 'with respect to x'

• The indefinite integral is defined by

  • $\int f\left(x\right)dx=F\left(x\right)+C$

    • where $F$ is a function such that $F^{\text{'}}\left(x\right)=f\left(x\right)$

      • $F$ is known as an antiderivative of $f$

    • and $C$ is any constant

      • $C$ is known as the constant of integration

• Integration is the inverse of differentiation

  • Integrating $f\left(x\right)$ gives you $F\left(x\right)$ ($+C$)

  • And differentiating $F\left(x\right)$ ($+C$) gives you $f\left(x\right)$

• Note that the indefinite integral of a function of $x$

  • is another function of $x$

Why do I need the constant of integration +C?

• To be an antiderivative of $f$, the function $F$ must satisfy $F^{\text{'}}\left(x\right)=f\left(x\right)$

• Say you found an $F\left(x\right)$ for which that is true

  • Add a constant to that

    • $F\left(x\right)+C$

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

    • $\frac{d}{dx}\left(F\left(x\right)+C\right)=F^{\text{'}}\left(x\right)+0=F^{\text{'}}\left(x\right)=f\left(x\right)$

  • I.e. if $F\left(x\right)$ is an antiderivative of $f\left(x\right)$

    • then $F\left(x\right)+C$ is also an antiderivative of $f\left(x\right)$

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

  • 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